Chasing a starting problem

the single faint "click" you hear is the starter relay closing. that shows neutral safety switch closed/operating properly.
If confirmed good battery connections to the starter doesn't cure it, a new starter will.
They do develop dead spots/poor connections inside.
I remember when dad would call me out to hold the key in start position while he bumped the starter with the tire tool. I recall a period of about 3 weeks when I beat the crap out of a starter on my Chevy LUV pickup. I was working 75 to 90 hours per week in a textile mill. Anyway...
Some modern mini starters ( like the one in my 96 Camry that is right there on the top, easily accessible ) have a yellow sticker on them that reads "DO NOT STRIKE WITH HAMMER". That suggests two things... striking to persuade a starter once was common practice, and these modern little starters are fragile. I mention this because lots of folks have put modern mini starters on these older engines.