Just west on highway 12 past the turn down to Bryce several dirt FS roads head off north. They're good roads, and after you drive a few miles its easy to find a little track heading off into the pines and a place to camp. Very convenient and pleasent.
There are lots of options for exploring and adding a day to the Egypt/Neon/25 mile loop. Dayhiking up along the bench between Neon and Choprock is pretty cool, as is hiking up the various branches of Choprock. Ringtail Canyon sould of Neon is worth exploring, and the various canyons both south and north of 25 mile are worthwhile, too. I'd hike down to the Escalante from the TH, set a camp, head up and explore Choprock, see Neon and Ringtail the next morning, camp somewhere in 25 mile, and have some time to wander that evening and the next morning before taking a wandering path back to the TH. The macro nav is easy on that route, while the micro stuff can be pretty challenging, so giving yourself time to backtrack and follow your nose is a good idea.
It'd be easy to spend three days exploring Coyote Gulch as well. Can't go wrong with either but IMO the first loop is more diverse.