I just saw your Jul 1998 posting.Our families may be connected. Our Allens, John and Margaret, were in Alabama in August, 1813. They had probably come to Alabama fromNorth Carolina through Georgia.John was Scottish descent. They were killed that October in a Creek Indian massacre at Ft. Mims. At the time they had two sons, Jesse Carter Allen (B. 1811) and James Madison Allen (b. 1799), the sons were not with them at Ft. Mims and survived. In the 1850 census, Jesse Carter Allen listed his birthplace as Mud Flats Tavern, Alabama.Both boys later moved to Santa Rosa County, Florida, where they are buried.
It seems likely that John Allen would have been in the same generation as your Jesse (possibly a brother) and could have named a son for him. Both our James Madisons were probably named for the Virginia governor and President James Madison, so that doesn't prove a lot.
If you do get any information on the parents of your Jesse Allen, please let me know. Thank you.