Quality Inn Downtown

Your options are quite limited

by PR-7

For better or worse, almost NO mountain areas in the U.S., and in particular those on federal land in the western U.S., have hiker huts like you find in Europe or in the northeast U.S. Here's info specific to the Superstition Mountains:

www.fs.fed.us/r3/tonto/wilderness/wilderness-superstition-index.shtml

Note that this area is legally designated wilderness, meaning significant improvements of ANY kind, definitely including covered lodging, are forbidden. Here's more info on what "wilderness" means:

www.fs.fed.us/r3/tonto/wilderness/wild_Faqs.shtml

Thus, to visit the Superstitions (which I HIGHLY recommend, if you are willing to plan intelligently and appropriately), you must stay in either a hotel (with day hiking) or a campground. Campgrounds, mostly with minimal facilities, are available in the Tonto National Forest:
www.fs.fed.us/r3/tonto/recreation/rec-camping-index.shtml
or you can do desert wilderness hiking (NOT for neophytes to the desert!).

To hike via hikers huts or hostels in the mountains, you'll probably have to visit the northeast U.S. -- which I wouldn't recommend in April for mountain hiking.