I no longer live in the Southern US but I am in southwestern Indiana and go to the Missouri and Arkansas Ozarks when I can.
When there is sufficient water, the Buffalo National River is about as good as it gets in the lower 48 states. I have been on the White River and Spring River. Have not yet been on the Mulberry or Cossatot. If you are looking for people to trip with, the Arkansas Canoe Club is a sizable group and they have a yearly Rendezvous, I believe on the Mulberry. You might check out their facebook page.
There is a group that gets together twice a year to paddle in the Missouri Ozarks, generally on the Jacks Fork River, Current River, North Fork of the White River, and Bryant's Creek. They generally do day trips but overnight and multi-day trips have also been done.
The Current and Eleven Point Rivers in Missouri are not too far away, can be paddled year round, and are suitable for multi-day trips. It is a bit further, but the Big South Fork of the Cumberland River flowing out of north Tennessee into Kentucky is a wonderful river for overnight trips as well.