Wade, I think you were born too late. Tennessee use to have numerous small clubs as well as some big ones that had competitive matches. As a matter of fact, Elk River, whom I am a member, was years ago very active and hosted the state championship with 100+ shooters being the norm. Elk River started it's downward slide after we quit hosting that big shoot. The decline and almost demise was when we were kicked off our leased range on nearby Cumberland Springs by the heirs of the Motlows who were obviously somewhat antigun.
Owl Hollow Gun Club near Franklin hosted the NMLRA Territorials and also The R. Southgate Memorial up until the owner of the range refused to renew their lease a few years back. Owl Hollow couldn't find any suitable land for a range and soon went belly up. My wife and myself had actually run a monthly match there for a few years and up until the club lost the range.
Others will have to tell you what happened to the Chickasaw Bluff Muzzleloaders down in Memphis or the John Sevier Longrifles in Knoxville. Neither have been active, as far as I know, in years.
The Tennessee Longhunters, I think, still have monthly or maybe quarterly shoots but I think they may be by invitation only to non members. I know a few of them hang out on this site and maybe they will chime in. If not, I can get you some contact info.
Your best bet will be in Alabama. I attended a silhouette shoot north of Birmingham this past weekend put on by The Alabama State Muzzle Loading Association, which is still quite viable. They have the NMLRA Territorial as well as the Alabama Championship at Brierfield Ironworks Park they also have monthly silhouette shoots at Magic City. If you are into shotguns the shoot in Clanton is the place to be.
The Georgia bunch is also still active but their shoots are a little more than a hike for you. At any event both the Alabama and Georgia folks are as fine as you will ever meet and I know most all of them. Contact me if you need any info as I don't mind helping you out if I can.