Hi Sebastian,
I’ll give you what I consider to be two really great options within your 3 hour restriction.
The first option is really from the Georgia North Carolina border to Wesser, NC, especially Albert Mountain, Standing Indian, Wayah Bald. You really can’t go wrong with that section. If you want to do a loop hike there are 2 places to manage rhe logistics: Standing Indian Campground in the Nantahala area, and the Nantahala Outdoor Center in Wesser, NC. The NOC will be crowded but food and (good) beer will be readily availabe before and or after your hike. Pick your 20 miles and have at it.
The other really outstanding option is to go a little bit further north to the southern end of the Great Smoky Mountain National Park. There are excellent opportunities there including the Appalachian Trail and loop options off of the Appalachian Trail like Gregory Bald. Call the ranger station and they can help you with your mileage…a really great bunch of folks.
I should also mention that shuttle options are fairly easy and reasonably priced if strictly AT is what you’re after. Good luck and have a great time.