2001 Escort poss starter problem

I have a 2001 Ford Escort ZX2. I live in northern Nevada in the high desert, and it gets pretty hot here - 104 today. Often when it’s really hot, I can’t get the car to start. All I get is a click. I’m guessing it’s the solenoid (which I guess you have to purchase with the starter), but I’d like to have someone else’s opinion. And I’m curious if it’s really heat related (it happened some last summer, but never happened over the winter or srping).

Can you use a multimeter and check the battery voltage? Can you check the voltage at the starter when the ignition key is turned turned, and held, at START?
How old is the battery? Have you ever removed the battery cables and cleaned them with a file, knife, or sandpaper?

I’m not sure how old the battery is. I bought the car a year ago I’ll start by cleaning the cables and terminals. Would that cause the car to refuse to start on an irregular basis? I’ll have to see if one of my buddies has a multimeter. I’m sure someone does - or I can take it into the auto hobby shop on base. I didn’t consider the battery because the car starts most of the time, and everything works (fan, radio, etc) even when it won’t start.

Thanks for the help.

Fords have, traditionally, had a starter solenoid on the inner fender; but, yours may not. My parts search at www.autozone didn’t reveal a starter solenoid; but, it did show a starter relay.
With the handy dandy multimeter, you could check the starter relay for power to the starter solenoid when the ignition switch is held at START. Actuate the ignition switch several times to see if the starter relay is energized each time.