Little River Chapter
Whitefeather - The Little River Chapter of TU works very closely with the GSMNP and is probably the one you are looking for. They were the primary organizing group for Troutfest (taking a hiatus this year, but should be back in 2014), and do a lot of work with the brook trout restoration, acid deposition, and other such efforts in the Park. Their website is http://lrctu.org and I'm sure they would be more than happy to welcome you as a new member. Of course, they also took me in as a member, so I don't really know what that says about their standards (or lack thereof).
Fly fishing - it's cheaper than a bass boat!