Maybe you might try tracing the J J Garner b. GA abt 1834 shown in the census, as it could be his brother? The marriage record provides no parents.
United States Census, 1860 Panola, Mississippi
Citing this Record: "United States Census, 1860," index, FamilySearch (https://familysearch.org/pal:/MM9.1.1/M6GD-S4M, Jackson Garner in household of Henry Garner, The Town Of Panola, Panola, Mississippi, United States; from "1860 U.S. Federal Census - Population," Fold3.com; p. 4, household ID 30, NARA microfilm publication M653; NARA microfilm publication M653. National Archives and Records Administration, Washington, D.C.; FHL microfilm 803,589.
Henry Garner M 20 Georgia
Mary B Garner F 20 Mississippi
Jackson Garner M 1 Mississippi
J J Garner M 24 Georgia
W M A Garner F 18 Tennessee
W P Garner M 0 Mississippi
F M Williams M 13 Mississippi
Mary Williams F 53 Tennessee
P G Williams F 10 Mississippi
United States Census, 1870 Caroline, Pulaski, AR
Citing this Record: "United States Census, 1870," index and images, FamilySearch (https://familysearch.org/pal:/MM9.1.1/MNCD-MRX, Henry Garner, Arkansas, United States; citing p. 53, family 392, NARA microfilm publication M593, National Archives and Records Administration, Washington D.C.; FHL microfilm 545,561.
Henry Garner M 35 Georgia
Mary Garner F 30 Tennessee
Jackson Garner M 12 Mississippi
James Garner M 9 Mississippi
Leonora Bell F 6 Arkansas
Name: Henry Garner
Spouse: Mary T. Bell
Marriage: 12 Jul 1857 Place: Tallahatchie, Mississippi
"Mississippi, Marriages, 1800-1911," index, FamilySearch (https://familysearch.org/pal:/MM9.1.1/V28V-RQS, Henry Garner and Mary T. Bell, 12 Jul 1857; citing Tallahatchie,Mississippi; FHL microfilm 894,879.