Chuck, I think the best way is to find all the information you can about him and, if that does not include his death information, it may at least help pinpoint some information that will help to narrow down your search. To that end, I found what may be his birth record at:
"Maine Births and Christenings, 1739-1900," database, FamilySearch (https://familysearch.org/ark:/61903/1:1:FWB7-KJC : 10 February 2018), James Gilmore, 15 Dec 1767; citing Woolwich, Sagadahoc, Maine; FHL microfilm 12,315.
This James Gilmore was born in Maine on 15 Dec 1767 and his parents were William and Elizabeth.