There is little to no service at any of the spots in the area (east outside of Capitol Reef), or at least that was the case for us in the fall of 2019.

Absolutely nothing in most of the park for cell signal. I went complete radio silent from my family for over 24 hours while were passed through for the night and then stayed to hike in the morning. Had to drive for quite a while out of the park before I was able to let everyone know I was still alive. Lol. I have T-mobile.