Well, there are tons of car clubs in east TN, especially Pontiac and Corvette it seems. Pigeon Forge's "strip" holds tons of huuuuuge car shows every year, including lots of club shows. There are also some in east Knox (at chilhowie Park) and in Morristown. And you're only a few hours from Charlotte, which holds a very large (thousands and thousands of people) car show twice(?) a year.
In short, I think you'd fit.

Most of the areas outside city limits have very few regulations so you'd not run afoul of the law by having parts cars sitting around in your yard or anything. You can live in the country and still be within 15-30 minutes of Chattanooga, Johnson City, and Knoxville.