Yes, in Zion, a lot!! But also on the right side of the Road (driving to Zion) Not in the NP But between Zion mountain ranch and the East Entrance Perhaps you can look at Alltrails. A good site for hikes

The dispersed camping area is pretty open so you can walk the area. However, we didn't really find any hiking trails. Our focus while staying here was to spend time in Zion which is pretty close. The hiking in Zion is almost limitless. We did Angels Landing and several others. Couldn't do any of the slot canyon hikes Zion is famous for, because they were all full of water (spring run off). You'll love it!

Zion is less than 13 miles away plenty of hiking trails. You can hike around this area must be careful no marked trails.