Then it is possibly the fan motor's resistance going up as it gets hot. You can remove the motor and lube the bearings (this works 50% of the time). I do not know if you can disassemble this motor or not, if you can. Then clean the commutator with emory cloth and use a pick to clean out any accumulation between the segments, also check and see if your brushes are not worn to the limit. You can get generic brushes at Ace Hardware (in the wall of many small drawers), and file them down to fit, they are soft carbon after all. You can also check the connectors on the speed resistors. There may be a bad solder joint or a corroded crimp, this increases the resistance when it gets hot.
You can also take the cover off then, take and hold a mirror so you can look at the squirrel cage. Turn off all the lights (in your garage) and run the motor. If the brushes are nearly gone, you will see some light from the gap the brushes have sparking if they are close to the limit.
Failing that, move to a warmer climate. My new truck's fan had never been turned on in a coon's age, it came from Phoenix. I had to reach in the back, turn the motor for a while and it started to turn, faster & faster, then has run fine since (along with a little Tri-Flow randomly squirted around inside it).