This time of year I would jet up to the Payson area and hit the Mogollon Rim. Probably a 1.5 hour drive, maybe a little more. Literally no end to beautiful remote dispersed camping spots, meadows, creeks, exc to explore.
I am not as familiar with stuff directly around the Phx area as I live in more northwestern AZ, but I have spent lots of time in the Camp Verde/Sedona/Oak Creek/Mogollon rim regions and it is phenomenal.
I know you're probably looking for stuff that is a little bit closer, but figured I would through that out there.