This 10,400 acre park is administered by the North Carolina Forest Service. Trails in this forest run from easy to difficult. The most popular is the Triple Falls Trail, which parallels the Little River and visits several waterfalls. Wading and swimming are allowed in season, as is camping with permits.
If you go:
The park is a little hard to find, but the website gives turn-by-turn directions from several nearby towns. The parking lots can get quite full on weekends, so anticipate a wait or a short hike to the trailheads.
Links:

204-210 Staton Road
Hendersonville, North Carolina 28739 USA
dupontsf.ncfs@ncagr.gov
Mailing address:
DuPont State Recreational Forest
PO Box 300
Cedar Mountain, NC 28718-0300 USA
Phone Numbers:
Dupont State Forest Service
+1 (828) 877-6527
+1 (828) 877-6527
Permits and Reservaations
+1 (828) 877-6527 ext. 242
+1 (828) 877-6527 ext. 242