starting problems

I have a 96 sc2 and the starting up process has been acting up lately. About a week ago i went to start it and it just clicked and i kept trying to turn it over and after about 20 times it started up very bad. After that it has happened about 5 times but I have noticed it only does it when my car is fully warmed up and when ive been driving for abuot more than an hour, not when its cold. I dont know if its a starter or what but I was hoping someone could help me out because all it does it click.( and its not my battery its brand new and I cleaned off the cables and everything) Thank you please help!!

I don't know about Saturn, but other GM starters that have a bad solenoid will seize up when they're hot.

just because it's a new battery doesn't mean it's good.Did the old battery start the same way? do the lights dim too much when trying to start?
I've found a hot engine needs more power to start than a cold one.
Also, I've had connections corrode at the starter motor. (on a different vehicle) that limits the amount of power delievered to the starter.