Camping within the Hot springs campground is literally right across the French Broad river, (a short hike) to gain access to the Appalachian trail, from there lovers leap is about a 10-20 min easy hike.
I think there is camping near Hot Springs but the trail itself is more of a quick day hike type escursion
Yes, the Hot Springs Campground is right on the trail because the trail is Bridge St. (the main street) in town. It is a great campground. Also on that street is Bluff Mountain Outfitters, a great place for hikers to restock/repair/recoup: https://g.co/kgs/ktKEzJ
There are backcountry sites along the trail. Several places with great vantage points overlooking the river (maybe not so great views when leaves are on the trees). Just continue on past Lovers Leap, gain a bit of elevation.
You see people camping alone the river, not sure if its allowed.
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