I'm in Memphis for at least a week every year and it's limited. I don't really climb so I can't help you specifically in that way, but I think you are going to need to head east. There is quite a bit of verticle rock as you get towards Nashville. I'm not positive, but I doubt there's much going on in the Meeman Shelby forest area along the shore of the old Miss, but you might have an interesting day out there if you just want to get out into the thick of it.
Tauk Sauk Mountain in Missouri and some of the mountains west of Little Rock. Woodall Mountain and Driskill if you are into state highpointing. Not exactly Seattle, Denver, Albany, or Salt Lake City but it looks like there are some opportunities.