Welcome - I'm sure you'll get lots of tips on places to go, so I won't try to list everything near Tennessee. However, I will mention one overlooked backpacking area: Mammoth Cave National Park. I know, the park's highlights are all underground. But, to preserve the underground part of the park, they had to buy up all the surface, too - and there's a lot of good backpacking on a well-developed set of trails. (Warning: some are hiking/biking trails.) No spectacular scenery, but just a good place to go be in the woods, with a number of pretty campsites and relatively abundant water.

Feel free to ask us any other questions you have, and again - welcome.