New starter, still problems.

I had been having started problems with my 3rd gen. It will start fine in the morning, but if I turn it off after driving a little, it won't want to turn over again for at least 10 minutes. It sounds like it's trying to engage the starter, but it's never "catching". After a few tries, I would here the starter just spinning. So, I replaced the starter this weekend. The shop said they had to drop the exhaust (which I knew that I wouldn't be able to do, so I took it to a shop) to access the starter and replace.
Now, I can tell there's a new starter because it engages very easily now, but I'm having the same problem when I turn it off, then try to start it up again. The starter doesn't spin, but it never "catches", ya know?
What could this be? Solenoid? And how much of a pain is that job??? I don't want to take it to the shop again if I don't have to, but I can't drop the exhaust myself.

You have headers?

new solenoid should have came on the starter. im guessing.

Nope. No headers.

Most likely a wire short then. I would check the condition of the wires running to the starter. The reason I asked about headers is cause under extreme heat the starter motor can freeze up or hesitate before it engages. I learned this after I installed some headers and tried to start my car after some long haul road trips.

Thanks for the lead, I'll check it out.