<?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>0001819252</rptOwnerCik>
            <rptOwnerName>Haymaker Sponsor III LLC</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>0</isDirector>
            <isOfficer>0</isOfficer>
            <isTenPercentOwner>1</isTenPercentOwner>
            <isOther>0</isOther>
        </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>
                    <footnoteId id="F1"/>
                </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>
                    <footnoteId id="F1"/>
                </underlyingSecurityShares>
            </underlyingSecurity>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>7937500</value>
                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>D</value>
                </directOrIndirectOwnership>
            </ownershipNature>
        </derivativeTransaction>
    </derivativeTable>

    <footnotes>
        <footnote id="F1">Steven J. Heyer, the Chief Executive Officer and a director of the issuer, and Andrew R. Heyer, the President and a director of the issuer, are the managing members of Haymaker Sponsor III LLC (the &quot;Sponsor&quot;) and have voting and investment discretion with respect to the securities held by the Sponsor. As such, Mr. Heyer and Mr. Heyer may be deemed to possess beneficial ownership of the securities held directly by the Sponsor. Mr. Heyer and Mr. Heyer disclaim any beneficial ownership of the reported securities other than to the extent of any pecuniary interest they 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&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 certain adjustments described therein and have 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 reporting person 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/ Steven J. Heyer
By Steven J. Heyer, Managing Member of Haymaker Sponsor III LLC</signatureName>
        <signatureDate>2021-03-09</signatureDate>
    </ownerSignature>
</ownershipDocument>
