SECURITIES AND EXCHANGE COMMISSION

WASHINGTON, DC 20549

SCHEDULE 13D/A

Under the Securities Exchange Act of 1934

(Amendment No. 1)*

Martin Midstream Partners L.P.

(Name of Issuer)

Common Units, representing limited partner interests

(Title of Class of Securities)

573331105

(CUSIP Number)

A. Wellford Tabor

Keeneland Capital, LLC

2820 Selwyn Avenue, Suite 550

Charlotte, North Carolina 28209

(704) 973-9001

Copy to:

Maynard, Cooper & Gale, P.C.

1901 Sixth Avenue North, Suite 2400

Birmingham, Alabama 35203-2618

(205) 254-1000

Attention: John P. Dulin, Jr.

(Name, Address and Telephone Number of Person Authorized to Receive Notices and Communications)

October 2, 2012

(Date of Event which Requires Filing of This Statement)

If the filing person has previously filed a statement on Schedule 13G to report the acquisition that is the subject of this Schedule 13D, and is filing this schedule because of §§240.13d-1(e), §§240.13d-1(f) or §§240.13d-1(g), check the following box  ¨.

Note: Schedules filed in paper format shall include a signed original and five copies of the schedule, including all exhibits. See Rule 13d-7 for other parties to whom copies are to be sent.

The information required on the remainder of this cover page shall not be deemed to be “filed” for the purpose of Section 18 of the Securities Exchange Act of 1934 (“Exchange Act”) or otherwise subject to the liabilities of that section of the Exchange Act but shall be subject to all other provisions of the Act (however, see the Notes).

This Amendment No. 1, filed with the Securities and Exchange Commission (the “Commission”) by KCM, LLC, a Delaware limited liability company (“KCM”), Keeneland Capital, LLC, a Delaware limited liability company (“Keeneland”), and A. Wellford Tabor, an individual resident of the State of North Carolina (“Mr. Tabor”), amends the Schedule 13D that was filed with the Commission on August 23, 2010 (the “Original Schedule 13D”, and together with Amendment No. 1, the statement on “Schedule 13D”) by KCM and Keeneland, relating to common units representing limited partner interests (“Common Units”) in Martin Midstream Partners L.P., a Delaware limited partnership (the “Issuer”). In this statement on Schedule 13D, KCM, Keeneland and Mr. Tabor are each referred to as a “Reporting Person” and are collectively referred to as the “Reporting Persons.” Any capitalized terms used and not defined herein shall have the meanings given to such terms in the Original Schedule 13D.

This statement on Schedule 13D relates to the Common Units of the Issuer. The address of the principal executive offices of the Issuer is 4200 Stone Road, Kilgore, TX 75662.

(a) This statement on Schedule 13D is jointly filed by the Reporting Persons. The Reporting Persons may be deemed to constitute a “group” for the purposes of Section 13(d)(3) of the Securities Exchange Act of 1934, as amended (the “Exchange Act”). The Reporting Persons have entered into a Joint Filing Agreement (a copy of which is filed with this statement as Exhibit 99.1 and is incorporated in this statement by reference) pursuant to which the Reporting Persons have agreed to file this statement jointly in accordance with the provisions of Rule 13d-1(k)(1) promulgated under the Exchange Act.

(b) The address of each of the Reporting Persons is c/o Keeneland Capital, LLC, 2820 Selwyn Avenue, Suite 550, Charlotte, North Carolina 28209.

(c) Mr. Tabor is the sole member of Keeneland, and Keeneland is the managing member of KCM and owns an interest in KCM. KCM and Keeneland are primarily involved in investment activities. The address of each Reporting Person is set forth in (b) above.

(d) and (e) None of the Reporting Persons during the last five years (i) has been convicted in a criminal proceeding (excluding traffic violations or similar misdemeanors) or (ii) has been a party to a civil proceeding of a judicial or administrative body of competent jurisdiction and as a result of such proceeding was or is subject to a judgment, decree or final order enjoining future violations of or mandating activities subject to, federal or state securities laws or finding any violation with respect to such laws.

(f) Mr. Tabor is a citizen of the United States. KCM and Keeneland are organized in the State of Delaware.

Pursuant to a Securities Repurchase and Settlement Agreement dated as of October 2, 2012 (the “Settlement Agreement”) by and among KCM, Scott D. Martin, an individual resident of the State of Texas, SKM Partnership, Ltd., a Texas limited partnership, and Martin Resource Management Corporation, a Texas corporation and the indirect owner of the Issuer’s general partner (“MRMC”), KCM acquired 1,500,000 Common Units that were held by a wholly-owned subsidiary of MRMC. In exchange for the Common Units and certain other consideration, including the settlement of certain litigation between the parties to the Settlement Agreement, KCM transferred to MRMC 1,423 shares of common stock of MRMC and an option to acquire an additional 3,049 shares of common stock of MRMC. As a result of the Settlement Agreement, the Reporting Persons no longer beneficially own any interest in MRMC.

The information set forth, or incorporated by reference, in Item 3 of this statement is hereby incorporated by reference.

The transfer of the 1,500,000 Common Units to KCM was registered by the Issuer under the Securities Act of 1933, as amended (the “Securities Act”), pursuant to a Registration Statement on Form S-3 filed by the Issuer with the Commission on August 22, 2012 (the “Registration Statement”). On October 2, 2012, the Issuer also filed with the Commission a prospectus supplement, which is a part of the Registration Statement, relating to such transfer.

The foregoing acquisition of Common Units was made for investment purposes in connection with the Settlement Agreement. KCM and MRMC have entered into a Lock-Up Agreement dated October 2, 2012 (the “Lock-Up Agreement”) whereby (i) until April 2, 2013, KCM has agreed not to offer or sell any of the Common Units without MRMC’s prior written consent, and (ii) for a period commencing on April 3, 2013 and ending on October 2, 2013, KCM has agreed not to sell more than 50,000 Common Units per week except for block trades. For purposes of the Lock-Up Agreement, the term “block trade” means any sale of 100,000 or more Common Units in a single transaction to a third party in a transaction not executed over a stock exchange so long as neither party makes any public announcement, filing or report unless required by law. The foregoing description of the Lock-Up Agreement is not complete and is qualified in its entirety by reference to the copy of the Lock-Up Agreement attached as Exhibit 99.2 hereto.

Except for a potential disposition of Common Units by KCM, none of the Reporting Persons nor, to the knowledge of the Reporting Persons, any other persons, has any current plans or proposals which relate to or would result in any transaction, event or action enumerated in paragraphs (a) through (j) of Item 4 of Schedule 13D promulgated under the Exchange Act.

Each of the Reporting Persons expects to evaluate on an ongoing basis the Issuer’s financial condition, business, operations and prospects, the market price of the Common Units, conditions in the securities markets generally, general economic and industry conditions and other factors. Accordingly, each Reporting Person reserves the right to change its plans and intentions at any time, as it deems appropriate. In particular, while the Reporting Persons do not presently intend to purchase any additional Common Units or other securities of the Issuer, the Reporting Persons (and their respective affiliates) may purchase additional Common Units or other securities of the Issuer. Additionally, except to the extent proscribed by the Lock-Up Agreement, the Reporting Persons may sell or transfer Common Units beneficially owned by them from time to time in public or private transactions. Any such transactions may be effected at any time or from time to time subject to any applicable limitations imposed on the sale of Issuer securities by the Lock-Up Agreement, the Securities Act and the rules and regulations promulgated thereunder or other applicable law.

The information set forth, or incorporated by reference, in Items 3, 4 and 6 of this statement is hereby incorporated by reference.

(a) Based on information reported by the Issuer in its Registration Statement on Form S-3 filed with the Commission on August 22, 2012, there were 23,116,776 Common Units outstanding as of August 22, 2012. The Reporting Persons own 1,500,000 Common Units of the Issuer, representing approximately 6.5% of the Issuer’s outstanding Common Units.

Mr. Tabor is the sole member of Keeneland, and Keeneland is the managing member of KCM and owns an interest in KCM. As a result, Mr. Tabor and Keeneland may be deemed to be the beneficial owner of the Common Units owned by KCM.

(b) The Reporting Persons have the power to vote or direct the vote and to dispose or to direct the disposition of the 1,500,000 Common Units beneficially owned by them.

(c) The information set forth or incorporated by reference, in Item 3 of this statement is hereby incorporated by reference.

(d) No other person is known by either Reporting Person to have the right to receive or the power to direct the receipt of dividends from, or the proceeds from the sale of, any Common Units owned by the Reporting Persons.

(e) Not applicable.

The information set forth or incorporated by reference, in Items 3, 4 and 5 of this statement is hereby incorporated by reference.

To the best of each Reporting Person’s knowledge, except as described in this Amendment No. 1, there are no contracts, arrangements, understandings or relationships among the persons named in Item 2 above, or between any such persons and any other person, with respect to any securities of the Issuer.

SIGNATURE

After reasonable inquiry and to the best of our knowledge and belief, the undersigned certify that the information set forth in this statement is true, complete and correct.

Date: October 2, 2012

EXHIBIT 99.1

JOINT FILING AGREEMENT

In accordance with Rule 13d-1(k)(1) promulgated under the Securities Exchange Act of 1934, as amended, the undersigned and each other person executing this joint filing agreement (this “Agreement”) agree as follows:

This Agreement may be executed in any number of counterparts, each of which shall be deemed to be an original, but all of which, taken together, shall constitute one and the same instrument.

In witness whereof, the undersigned have executed this Agreement as of October 2, 2012.

EXHIBIT 99.2

LOCK-UP AGREEMENT

This Lock-Up Agreement (this “Agreement”) is made and entered into as of the 2nd day of October, 2012, by and between KCM, LLC, a Delaware limited liability company (“KCM”), and Martin Resource Management Corporation, a Texas corporation (the “Company”). Except as otherwise indicated herein, capitalized terms used but not otherwise defined herein shall have the meanings ascribed to such terms in that certain Securities Repurchase and Settlement Agreement, dated October 2, 2012 (the “Repurchase and Settlement Agreement”), among KCM, the Company, Scott D. Martin, an individual resident of the State of Texas (“Martin”), and SKM Partnership, Ltd., a Texas limited partnership (“SKM”; and together with Martin, the “Martin Parties”).

WHEREAS, KCM, the Company and the Martin Parties have entered into the Repurchase and Settlement Agreement, which provides for, among other things, the conveyance by the Company or an Affiliate of the Company to KCM or its designated Affiliate, on the Closing Date, free and clear of all Encumbrances, One Million Five Hundred Thousand (1,500,000) common units of limited partnership interests (the “Common Units”) of Martin Midstream Partners, L.P., a Delaware limited partnership (“MMLP”); and

WHEREAS, the Repurchase and Settlement Agreement provides that KCM shall have the right and ability to sell or transfer the Common Units, at any time and from time to time following the Closing Date, without any restrictions or limitations other than pursuant to restrictions set forth in this Agreement; and

WHEREAS, in order to induce each of KCM and the Company to enter into the Repurchase and Settlement Agreement and to induce the Company (or an Affiliate of the Company) to convey the Common Units to KCM and to induce KCM to accept the Common Units from the Company (or an Affiliate of the Company) pursuant to the terms of the Repurchase and Settlement Agreement, KCM and the Company are entering into this Agreement.

NOW, THEREFORE, in consideration of the transactions provided for in the Repurchase and Settlement Agreement and the mutual promises and covenants contained herein, the parties hereto agree as follows:

1. Resale Restrictions.

(a) KCM agrees that, during the period commencing on the Closing Date and ending on the date that is six months after the Closing Date (the “Lock-Up Period”), KCM will not offer, sell, contract to sell, pledge, grant any option or contract to purchase, purchase any option or contract to sell, or otherwise transfer or dispose of, directly or indirectly, any of the Common Units, or any securities convertible into or exercisable or exchangeable for the Common Units, unless KCM has obtained the Company’s prior written consent; provided, that the restrictions set forth in this Section 1(a) shall not apply to any sale or other transfer of the Common Units to (i) the Company or any of its Affiliates (including MMLP) or (ii) any Affiliate of KCM in which such Affiliate agrees in writing prior to the transfer to be bound by the provisions of this Agreement as fully as KCM.

(b) KCM agrees that, for a period commencing on the first day following the expiration of the Lock-Up Period and ending on the one-year anniversary of the Closing Date, KCM will not sell more than 50,000 Common Units per week except for Block Trades; provided, that the restrictions set forth in this Section 1(b) shall not apply to any sale or other transfer of the Common

Units to (i) the Company or any of its Affiliates (including MMLP) or (ii) any Affiliate of KCM in which such Affiliate agrees in writing prior to the transfer to be bound by the provisions of this Agreement as fully as KCM. For purposes of this Agreement, the term “Block Trade” shall mean any sale of 100,000 or more Common Units in a single transaction to a third party in a transaction not executed over a stock exchange so long as neither party makes any public announcement, filing or report unless required by law. In the case of a Block Trade of 300,000 or more Common Units to any one third party (or any Affiliate of such third party), the purchaser must agree in writing to be bound by the provisions of this Agreement (excluding the provisions of Section 1(a) and Section 2).

(c) KCM agrees that, without the Company’s prior written consent, neither KCM nor any Affiliate of KCM shall knowingly offer, sell, contract to sell, pledge, grant any option or contract to purchase, purchase any option or contract to sell, or otherwise transfer, directly or indirectly, any of the Common Units, or any securities convertible into or exercisable or exchangeable for the Common Units, to the Martin Parties, whether during or after the Lock-Up Period.

(d) The restrictions on transfer set forth in Sections 1(a) and 1(b) shall terminate immediately and be of no further force or effect upon the earliest to occur of (i) a breach by the Company of any its representations, warranties, covenants, obligations or agreements in this Agreement or the Repurchase and Settlement Agreement, provided that KCM shall give written notice of such breach to the Company and the Company shall have ten days to cure such breach, (ii) the failure by MMLP to timely file with the SEC, pursuant to SEC rules promulgated under the Exchange Act, any of its periodic reports on Forms 10-Q or 10-K, subject to the filing extensions provided for in Rule 12b-25 under the Exchange Act, (iii) the occurrence of a Change in Control Event; provided that, for purposes of the definitions of “Change in Control Event” and “Incumbent Holders” as used or interpreted in this Section 1(d), the references to the “Company” shall be deemed to be references to either “MMLP” or the “Company” and shall not refer exclusively to a Change in Control Event involving the Company, (iv) the delisting of MMLP’s common units from the Nasdaq Stock Market or any other national securities exchange on which such common units are listed, (v) the sale or transfer by MMLP, directly or indirectly, on or after the Closing Date of more than 4,000,000 common units of MMLP, or securities convertible into or exercisable or exchangeable for more than 4,000,000 common units of MMLP, in a capital raising transaction, or (vi) any sale or transfer, or the entry into a contract for the sale or transfer, of any of the MRMC Units, or any securities convertible into or exercisable or exchangeable for the MRMC Units, on or after the Closing Date, except for sales or transfers to and among the Company and Affiliates of the Company. “MRMC Units” means the common units of MMLP that are owned by the Company or any of its subsidiaries (excluding common units held in treasury by MMLP) as of the Closing Date, but excluding the Common Units to be transferred by the Company to KCM at Closing. Furthermore, notwithstanding anything in this Agreement to the contrary, the restrictions on transfer set forth in Sections 1(a) and 1(b) shall not apply to any sale of the Common Units in an Underwritten Offering.

(e) KCM agrees and consents to the entry of stop transfer instructions with MMLP’s transfer agent and registrar relating to the transfer of the Common Units by KCM except in compliance with the restrictions described in this Section 1. Promptly following the expiration or early termination of the Lock-Up Period, the Company agrees to cause MMLP’s transfer agent and registrar to remove any transfer restrictions on the Common Units and any restrictive legends that appear on the KCM Certificate as soon as is reasonably practicable. The “KCM Certificate” means the common unit certificate that is issued to KCM at or promptly following Closing in exchange for the Common Unit Certificate pursuant to Section 1.4(a) of the Repurchase and Settlement Agreement, as well as any common unit certificate that is subsequently issued in exchange therefor.

2. Indemnification.

(a) The Company will indemnify and hold harmless KCM, each of KCM’s Affiliates, and each of their respective employees, agents and representatives (each, a “Covered Party”) against any and all losses, claims, damages, expenses or liabilities, joint or several, to which such Covered Party may become subject, under the Securities Act, the Exchange Act or otherwise, including any amounts paid in settlement of any litigation, or any investigation or proceeding by any governmental agency or body, commenced or threatened (“Losses”), insofar as such Losses (or actions in respect thereof) arise out of or are based upon an untrue statement or alleged untrue statement of a material fact contained in the S-3 Registration Statement, the Prospectus Supplement, or any amendment or supplement thereto, any “issuer free writing prospectus” (as defined in Rule 433 under the Securities Act) that is filed or required to be filed by MMLP with the SEC, or any other document filed or required to be filed by MMLP with the SEC pursuant to the Securities Act or the Exchange Act that is incorporated, or is required to be incorporated, by reference into the S 3 Registration Statement pursuant to Item 12 of Form S-3, or arise out of or are based upon the omission or alleged omission to state therein a material fact required to be stated therein or necessary to make the statements therein not misleading, and will reimburse each Covered Party for any legal or other expenses reasonably incurred by such Covered Party in connection with investigating, preparing or defending any such action or claim as such expenses are incurred; provided, however, that the Company shall not be liable in any such case to the extent that any such Loss arises out of or is based upon an untrue statement or alleged untrue statement or omission or alleged omission made in the Prospectus Supplement in reliance upon and in conformity with written information furnished to the Company by KCM expressly for use therein.

(b) KCM will indemnify and hold harmless the Company and each of its Affiliates (each a “Company Covered Party”) against any and all Losses to which such Company Covered Party may become subject, under the Securities Act or otherwise, insofar as such Losses (or actions in respect thereof) arise out of or are based upon an untrue statement or alleged untrue statement of a material fact contained in the Prospectus Supplement, or arise out of or are based upon the omission or alleged omission to state therein a material fact required to be stated therein or necessary to make the statements therein not misleading, in each case to the extent, but only to the extent, that such untrue statement or alleged untrue statement or omission or alleged omission was made in the Prospectus Supplement in reliance upon and in conformity with written information furnished to the Company by KCM expressly for use therein; and will reimburse the Company Covered Party for any legal or other expenses reasonably incurred by such Company Covered Party in connection with investigating or defending any such action or claim as such expenses are incurred.

3. Governing Law. This Agreement shall be governed by and construed in accordance with the laws of the State of Texas without regard to conflict of law principles.

4. Counterparts. This Agreement may be executed in counterparts, each of which shall be deemed to be an original, but all of which together shall constitute one and the same document with the same force and effect as though both parties to this Agreement had executed the same document. The exchange of copies of this Agreement and of signature pages by facsimile transmission or electronic copies (such as .pdf copies) shall constitute effective execution and delivery of this Agreement as to the parties hereto and may be used in lieu of the original Agreement for all purposes. Signatures of the parties hereto transmitted by facsimile or electronic mail shall be deemed to be the original signatures for all purposes.

5. Captions; Articles and Sections; Construction. The captions contained in this Agreement are for reference purposes only and are not part of this Agreement. Unless otherwise indicated, all references to particular Sections shall mean and refer to the referenced Sections of this Agreement. Where the context so requires or permits, the use of singular form includes the plural, and the use of the plural form includes the singular, and the use of any gender includes any and all genders. The words “hereof,” “herein,” and terms of similar import shall refer to this entire Agreement. Unless the context clearly requires otherwise, the use of the terms “including,” “included,” “such as,” or terms of similar meaning, shall not be construed to imply the exclusion of any other particular elements.

6. Waiver of Jury Trial. TO THE FULLEST EXTENT PERMITTED BY APPLICABLE LAW, EACH PARTY TO THIS AGREEMENT, BY EXECUTION HEREOF, KNOWINGLY, VOLUNTARILY, AND INTENTIONALLY WAIVES ANY RIGHT IT MAY HAVE TO A TRIAL BY JURY IN RESPECT OF ANY LITIGATION TO ENFORCE THIS AGREEMENT OR SEEK A REMEDY RESULTING FROM A BREACH OF THIS AGREEMENT. THIS PROVISION IS A MATERIAL INDUCEMENT TO THE PARTIES TO ENTER INTO AND ACCEPT THIS AGREEMENT.

7. Entire Agreement. This Agreement, together with the Repurchase and Settlement Agreement (including the schedules and exhibits attached thereto, and including the documents, agreements and instruments referred to therein or executed in connection with the Repurchase and Settlement Agreement), and the transactions provided for herein and therein constitute the entire agreement between the parties hereto with respect to the transactions contemplated hereunder and thereunder and supersede all prior arrangements or understandings with respect thereto, written or oral. No covenant or condition not expressed or contemplated in this Agreement or the Repurchase and Settlement Agreement shall affect or be effective to interpret, change or restrict this Agreement. Any term of this Agreement may be amended only with the mutual written consent of each of the parties hereto.

[Signature page follows]

IN WITNESS WHEREOF, this Agreement has been duly executed and delivered by the undersigned as of the date and year first above written.

SIGNATURE PAGE TO LOCK-UP AGREEMENT

SIGNATURE PAGE TO LOCK-UP AGREEMENT