[tbracker]

I had a very similar problem with my Everett-Morrison with Genesis alum. 427FE motor. Very often it would just click, usually when it was hot. I'd rock the car and sometimes this worked but mostly not. I was told by someone that the started was out of alignment needed to be shimmed. This just didn't seem to make sense so I tried other things.
I took the starter to a shop and had it tested...OK.
I put insulation on the starter...no difference.
I talked to a reliable engine guy who said these motors don't need to be shimmed check the voltage.
I put a meter on the starter and found I wasn't getting as much voltage at the solenoid as I should have. I checked the voltage at various points in the starter circuit and found I was loosing a fair amount between the battery, key, solenoid routing. I put a simple relay in with the ignition switch closing the relay and battery juice then going straight to the starter solenoid.
It has worked like a champ ever since.
I don't know if your car employs a starter relay or just uses the ignition switch to crank the starter as mine did but this mod worked for me.