This is a road that runs next to Capitol Reef NP. Has a few BLM roads that run off of this along with hiking areas. No designated campgrounds unless your staying in the park. you could try boondocking but understand there are private ranches along this road. I was there in 2016. I don't remember any cell reception outside of Torrey and a few spots in the park.
You might try a mile or so east on State Route 24. E Hartnet Rd S. Not sure if you want the road noise but you are right next to the river.