Car Camping Near Clay Hills Takeout after San Juan Trip

Does anyone know of good spots to car camp after taking out at Clay Hills? We would rather not camp at the ramp. It looks like Natural Bridges National Monument could be an option and is under 2 hours away. However, it is first come first serve, so no idea if there would be room for us and we probably won't arrive until later in the day. We're totally fine with dispersed camping too, but it would be nice to have a plan vs. aimlessly driving looking for something that would work. Are there...

Car Camping Near Clay Hills Takeout after San Juan Trip