Tom, It appears that "Inspiration" was indeed a town, although it sounds like more of a mining camp to me because there's no record I can find of a post office there.
This is from "ARIZONA PLACE NAMES," by Will C. Barnes;
"Mining town and important copper camp about 3 miles west of Globe. "At one time the owners, being in hard straits for money, had the "inspiration" to borrow money from a bank. Being successful they called the mine "Inspiration," after the thought."
You might stop at the Mining Museum across from the Old Dominion Mine in Globe and check their archives for further information and maybe even photographs of the smokestack and boiler plant being built. Donna Anderson who volunteers at the museum would be a great person to contact, she's written extensively on the area and is very knowledgeable.
You might also be interested in seeing this, you might find some pertinent dates here;