NC State parks allow hanging at all camp sites unless specifically stated otherwise. Oddly enough though it is practically impossible to hang at Hammocks State Park due to lack of suitable trees, so I've been told
The only trees I saw suitable for hanging were around the park HQ (on the mainland) where camping is not allowed. Out on the island where camping is allowed I did not see any trees you could hang from. If have a kayak or boat you might be able to find a place to hang on one of the many small islands close to the island part of the park.
I had a hammock in my day pack in case I found a spot but never did