Your Jesse was the son of Tillman Satterfield and was likely named after his uncle, who was likely named after his uncle (my Jesse Franklin). Below is the 1880 Lumpkin County, Georgia census entry for Tillman.
Satterfield T.L. (M) 32 GA, Josephine (F) 28 GA, Noah (M) 9 GA, Jessey (M) 7 GA, William (M) 4 GA
I looked through the rest of the 1880 Georgia Satterfield entries and didn't see another Jesse close to being the right age. Tillman was the son of John Satterfield and Aldelphia Trammell. John was the first son of James D. Satterfield and Sarah Corbin. Sarah Was the daughter of Peter and Elizabeth Corbin. James D. was the son of James W. Satterfield and Elizabeth (unknown but possibly Bridges, Moore, or Wilson). Dot Flanagan is descended from Tillman's brother Jesse and has done more research on this branch of the family than I have. She can be reached at email@example.com