<?xml version="1.0"?>
<ownershipDocument>

    <schemaVersion>X0306</schemaVersion>

    <documentType>4</documentType>

    <periodOfReport>2021-03-05</periodOfReport>

    <notSubjectToSection16>0</notSubjectToSection16>

    <issuer>
        <issuerCik>0001819253</issuerCik>
        <issuerName>Haymaker Acquisition Corp. III</issuerName>
        <issuerTradingSymbol>HYACU</issuerTradingSymbol>
    </issuer>

    <reportingOwner>
        <reportingOwnerId>
            <rptOwnerCik>0001259062</rptOwnerCik>
            <rptOwnerName>HEYER ANDREW R</rptOwnerName>
        </reportingOwnerId>
        <reportingOwnerAddress>
            <rptOwnerStreet1>501 MADISON AVENUE, FLOOR 12</rptOwnerStreet1>
            <rptOwnerStreet2></rptOwnerStreet2>
            <rptOwnerCity>NEW YORK</rptOwnerCity>
            <rptOwnerState>NY</rptOwnerState>
            <rptOwnerZipCode>10022</rptOwnerZipCode>
            <rptOwnerStateDescription></rptOwnerStateDescription>
        </reportingOwnerAddress>
        <reportingOwnerRelationship>
            <isDirector>1</isDirector>
            <isOfficer>1</isOfficer>
            <isTenPercentOwner>0</isTenPercentOwner>
            <isOther>0</isOther>
            <officerTitle>President</officerTitle>
        </reportingOwnerRelationship>
    </reportingOwner>

    <derivativeTable>
        <derivativeTransaction>
            <securityTitle>
                <value>Class B Common Stock</value>
            </securityTitle>
            <conversionOrExercisePrice>
                <footnoteId id="F2"/>
            </conversionOrExercisePrice>
            <transactionDate>
                <value>2021-03-05</value>
            </transactionDate>
            <deemedExecutionDate></deemedExecutionDate>
            <transactionCoding>
                <transactionFormType>4</transactionFormType>
                <transactionCode>J</transactionCode>
                <equitySwapInvolved>0</equitySwapInvolved>
                <footnoteId id="F3"/>
            </transactionCoding>
            <transactionTimeliness>
                <value></value>
            </transactionTimeliness>
            <transactionAmounts>
                <transactionShares>
                    <value>687500</value>
                </transactionShares>
                <transactionPricePerShare>
                    <value>0.00</value>
                </transactionPricePerShare>
                <transactionAcquiredDisposedCode>
                    <value>D</value>
                </transactionAcquiredDisposedCode>
            </transactionAmounts>
            <exerciseDate>
                <footnoteId id="F2"/>
            </exerciseDate>
            <expirationDate>
                <footnoteId id="F2"/>
            </expirationDate>
            <underlyingSecurity>
                <underlyingSecurityTitle>
                    <value>Class A Common Stock</value>
                </underlyingSecurityTitle>
                <underlyingSecurityShares>
                    <value>687500</value>
                </underlyingSecurityShares>
            </underlyingSecurity>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>7937500</value>
                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>I</value>
                </directOrIndirectOwnership>
                <natureOfOwnership>
                    <value>See footnote</value>
                    <footnoteId id="F1"/>
                </natureOfOwnership>
            </ownershipNature>
        </derivativeTransaction>
    </derivativeTable>

    <footnotes>
        <footnote id="F1">Andrew R. Heyer, the President and a director of the issuer, is a managing member of Haymaker Sponsor III LLC (the &quot;Sponsor&quot;) and has voting and investment discretion with respect to the securities held by the Sponsor. As such, Mr. Heyer may be deemed to possess beneficial ownership of the securities held directly by the Sponsor. Mr. Heyer disclaims any beneficial ownership of the reported securities other than to the extent of any pecuniary interest he may have therein, directly or indirectly</footnote>
        <footnote id="F2">As described in the issuer's registration statement on Form S-1 ((File No. 333-253010) under the heading &quot;Description of Securities--Founder Shares and Placement Shares&quot;, the shares of Class B common stock will automatically convert into shares of Class A common stock at the time of the issuer's initial business combination, on a one-for-one basis, subject to adjustment for stock splits, stock dividends, reorganizations, recapitalizations and the like, and certain anti-dilution rights and has no expiration date.</footnote>
        <footnote id="F3">As contemplated in connection with the initial public offering of the issuer, 687,500 shares of Class B common stock of the issuer were returned by the Sponsor to the issuer for no consideration and cancelled because the underwriters' over-allotment option was not exercised in full.</footnote>
    </footnotes>

    <remarks></remarks>

    <ownerSignature>
        <signatureName>/s/ Andrew R. Heyer</signatureName>
        <signatureDate>2021-03-09</signatureDate>
    </ownerSignature>
</ownershipDocument>
