In terms of Grand Staircase/Escalante, try heading down Hole in the Rock Road. There are spur roads off the main road where you'll find camping spots. There are a variety of interesting hikes in this area, including Peekaboo and Spooky. If there are gates, just park and walk. Check out
this webpage for info on various hikes in this area (Listed under Escalante River). You probably can't get into Lower Calf Creek Falls, but most of the rest might be accessible, even in you have to walk a little further because of a closed gate (I just don't remember if there are gates near the trailheads or not).
