Do you know his date of death? An obituary would be a good step if you haven't found one already.
There are a lot of Carters in Tate at the time he was there, including some born in Ohio in the early 1800s. There is a George Carter born 1807 who doesn't look like he could be Phineas's brother -- there are many public member trees of him on Ancestry and his siblings seem pretty filled in but some of the better trees seem to have cousins etc. Just make sure there are actually sources and not just pasted-together trees. In general, I've been helped a lot by keeping an eye on neighbors with similar names so that I recognize them when a reliable record appears.
If you have any clues as to whether Phineas was religious, and if so what sect, finding a church in Tate may be helpful. If any of the other Carters are relatives, they could be witnesses to any baptisms of his children.
Also, have you looked at the historical society websites for the counties he lived in? He may appear in a book of local biographical sketches or something along those lines.