Hi,

I have a rebuilt original starter which works fine when the engine is relatively cold (assuming at that point starter is also less hot!!!).

If I turn off the engine at operating temp. and try to start it again my starter just clicks. It seems that plunger when hot has some issues. It happened to me the other day and I waited for everything to cool down for 20 mins and started the car again fine. I dont believe it is the inhibitor switch etc.

I have a rebuilt starter on a different classic car and I am seeing similar behavior. Anyone seen similar issue?

Check the voltage at the solenoid when trying to start. It would be nice to know the current flow into the solenoid, too.
The other option would be to throw a new solenoid at it, but I'm not a fan of the parts cannon.