In the 1930 and 1940 Census, he's living with George Hamilton because his mother remarried.
Homer was born about 1900. He is the son of Unknown Taylor.
Sources
"United States Census, 1910," database with images, FamilySearch (https://familysearch.org/ark:/61903/1:1:M28Y-SP9 : accessed 24 May 2017), Homer Taylor in household of James Taylor, Magisterial District 3, Daviess, Kentucky, United States; citing enumeration district (ED) ED 39, sheet 10A, family 185, NARA microfilm publication T624 (Washington D.C.: National Archives and Records Administration, 1982), roll 473; FHL microfilm 1,374,486.
"United States Census, 1920," database with images, FamilySearch (https://familysearch.org/ark:/61903/1:1:MHLQ-FB5 : accessed 24 May 2017), Leona Taylor in household of Homer Taylor, Sorgho, Daviess, Kentucky, United States; citing ED 46, sheet 4B, line 53, family 72, NARA microfilm publication T625 (Washington D.C.: National Archives and Records Administration, 1992), roll 567; FHL microfilm 1,820,567.
"United States Census, 1930," database with images, FamilySearch (https://familysearch.org/ark:/61903/1:1:XM6Z-GVW : accessed 24 May 2017), Jeffie Taylor in household of George Hamilton, District 3, Daviess, Kentucky, United States; citing enumeration district (ED) ED 32, sheet 2B, line 93, family 40, NARA microfilm publication T626 (Washington D.C.: National Archives and Records Administration, 2002), roll 741; FHL microfilm 2,340,476.
"United States Census, 1940," database with images, FamilySearch (https://familysearch.org/ark:/61903/1:1:K7RJ-S1X : accessed 24 May 2017), Jeff Taylor in household of George Hamilton, Owensboro, Magisterial District 1, Daviess, Kentucky, United States; citing enumeration district (ED) 30-5, sheet 18A, line 28, family 385, Sixteenth Census of the United States, 1940, NARA digital publication T627. Records of the Bureau of the Census, 1790 - 2007, RG 29. Washington, D.C.: National Archives and Records Administration, 2012, roll 1298.
"United States Social Security Death Index," database, FamilySearch (https://familysearch.org/ark:/61903/1:1:JR5Y-H32 : 20 May 2014), Homer Taylor, 10 Feb 1989; citing U.S. Social Security Administration, Death Master File, database (Alexandria, Virginia: National Technical Information Service, ongoing).
"Kentucky, Vital Record Indexes, 1911-1999," database, FamilySearch (https://familysearch.org/ark:/61903/1:1:QKH1-CJH3 : 1 July 2015), Homer L Taylor, 10 Feb 1989; citing Death, Daviess, Kentucky, United States, Kentucky Department for Libraries and Archives, Frankfort.
"United States, GenealogyBank Obituaries, 1980-2014," database with images, FamilySearch (https://familysearch.org/ark:/61903/1:1:QVPL-DMDH : 13 September 2016), Homer Lee Taylor Jr, Kentucky, United States, 11 Feb 1989; from "Recent Newspaper Obituaries (1977 - Today)," database, GenealogyBank.com (http://www.genealogybank.com : 2014); citing Owensboro Messenger-Inquirer, born-digital text.
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DNA Connections
It may be possible to confirm family relationships with Homer by comparing test results with other carriers of his ancestors' Y-chromosome or mitochondrial DNA.
However, there are no known yDNA or mtDNA test-takers in his direct paternal or maternal line.
It is likely that these autosomal DNA test-takers will share some percentage of DNA with Homer: