There are so many options in that area. And April can be a great time to hike. The wildflowers will be out. You’ll likely get rain, though.
There are lots of AT hikes. My AT favorites would include Grayson Highlands or Roan Highlands if you’ve never been.
The Smokies are an option. There are good loops with easy access out of the Big Creek trailhead. Reservations required.
Slickrock WIlderness just southeast of the Smokies is a good option. No permits needed, no crowds, good loop options, and old growth forest.
Big Frog/Cohutta Wildernesses look like a good option. I have not been yet but hope to go soon.