I don't have much advice for Chattanooga proper, but you could always try Lafayette, GA, which isn't too far south. I fly over there regularly, and *think* you could find hangar space there. It seems to be an aviation friendly place.
Another airport just south of you is Dalton, GA. It is a very underutilized GA field. The Airport Manager, who also runs the FBO has created multiple policies that make the airport very unfriendly to GA. Nice runway, plenty of ramp and hangar space, but no airplanes. There is a reason...