In the 1850 McMinn Co., TN Census;
# 1168
Susan CARTER age 75 and Martha age 25 are next door to J.T. Carter.
In the Wills and Estates for McMinn Co. there is a listing for Jessie Carter
WB E 82-84, 25 Jan 1850. Acct. of sale and Invt. by James Sewell, Adm.; personal property in Monroe Co.
WB E 88, 11 Jan 1850. Comm. John Neill, Philip Fry, and D.F. Jameson set apart year's provisions for widow Susan and family.
WB E 197, 10 Mar 1852. Sett. by same Adm.; paid equal amounts to heirs Peter, James, Henderson, John, and Lewis M. Carter, Margaret Trim, and Levi Pressley.

(I have no proof that this is your J.T. Carter family. But there was a Susan listed next door in 1850. So it gives you some people to check out.)
If you wish to get a complete copy of the Court Action listed above, write to;(they will charge a fee to make copies)
County Court Records
Wills and Estates
McMinn Co. Courthouse
Athens, TN 37303