I have a '88 Cutass Supreme that is hard to start when its hot. Does it need a new starter?

Can you clarify "hard to start?" Does it crank the same when it is hot or cold, or is it not cranking when the engine gets hot? The starter may be too close to the exhaust. If that's the case, you can either replace the starter with an aftermarket high-performance starter from any speed parts catalog which will be smaller and thus further from the exhaust, or you can wrap the exhaust header with heat wrap tape, also from any speed parts catalog. If, on the other hand, it cranks the same hot or cold, but it's not starting when hot, it may be something to do with the fuel delivery, like vapor lock.

Maybe I did not explain well. The starter makes no engage click or sound. The only way I have found to start the car is to go the solenoid and jump the starter. To me that sounds like a bad place on the starter's coil, does that sound right? Something is not engaging in the starter itself. You're right the starter in this car is close to the exhaust manifold (its front wheel drive) and after it sits the manifold may be radiating heat to the starter? Would a new starter fix the issue? This one is about 5 years old.

A starter should last more than 5 years, but the symptoms you describe do sound like a starter issue. I wouldn't worry too much about its proximity to the exhaust. It worked fine for the last 28 years so there's no reason to think its suddenly getting too hot.