Jones Gap would be close to the top of my list. If you can get site 6 or 7 on the office spur, they're close enough to the parking lot to carry extra gear... table, chair, stove, lantern, cooler, etc.
Here are a couple of trip reports from site #6. I have hung in three different places on site 6.
Paris Mountain is in the Greenville city limits so it's a little closer than Jones Gap. You can hike about an hour to one of the five primitive sites on the North Lake Loop. Reservation required.
All have hang opportunities.
Here is a pic of my favorite, site #4.
The Chattooga River is the crown jewel of this area though. Multiple hanging opportunities for groups or singles. It'll be a two hour drive from Piedmont to the trailhead but you won't be disappointed.
I haven't seen anything in this thread that tells the date. Did I miss it?