From the little experience I have on the subject I would say Hiking in above Elkmont would be a great option. Unfortunately campsite #24 has been closed due to bear activity. You could hike further up to campsite #30, but the water might be too skinny to fish (Although its been pouring here in buckets lately...)
Campsite #18 on the West Prong of the LR is another option assuming that the water has come up some.
The NC side has some good options as well. If you want to hike a ways, you could hike up Eagle Creek from the Fontana Dam and stop at a few different sites. Also, I believe you could take the "road to nowhere" outside of Bryson City to Noland Creek and hike in from there. There are a few site on that stream as well.
There are some folks on the board that do a lot of hiking/fishing, I'm looking forward to their suggestions...This is a subject that I'd really like to learn a lot more about as well.
Does anyone know of a guidebook to the backcountry campsites in the Smokies? I've found limited info from different sources, but nothing that really covers all of the backcountry sites. Most of the "tent" camping books deal with the fee-area sites and not the backcountry sites. John Malloy has written a book that describes a few of the backcountry sites, but only lightly, and only as part of overnight loop hikes.
My sons and I hiked into and camped at #20 a few weeks back and never saw another soul. It was a great site (unfortunately not close to a fishable stream), but we are always looking for great backcountry adventures...