That's a tough one...we usually hike to hang. Try Goldhead State Park. It's a nice, quiet, remote old park, and I heard they are hammock friendly. We've stayed there many times, but have not hammocked it yet.
I've also heard good things about Cary State Forest:
Something hidden. Go and find it. Go, and look behind the Ranges. Something lost behind the Ranges. Lost and waiting for you . . . Go! - Rudyard Kipling