95 grand am will not turn over

1995 Pontiac Grand Am
V6 3.1 liter with 175,000 miles

The temps were around 35F and the car was starting just fine. The temps dropped to 15F and then colder and the car wouldn't start at all. When I turn the key/ignition over all of the lights on the dash come on, but the engine doesn't turn over and I here no clicking. The first time that I tried starting the car after the temp had dropped I pulled the key out of the ignition and the engine turned over on its own (without the key). I had to put the key back in the ignition to disengage the starting system. Since then it won't turn over at all. I did replace the battery since it was much needed anyways, but the problem remains. With the new battery, when trying to start the car I do hear faint moaning sound possibly coming from the starter motor or solenoid (not familiar), but the engine still doesn't even turn over. Does anybody have any ideas as to what this could be? Thanks.

Starter has moisture in it and has frozen up. Happens all the time here in arkansas during the winter. Get a block heater. Problem solved. Next!
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hit the starter with broom stick it will start. Sounds like time for new starter motor or solenoid.

Thanks! Great ideas! I did try hitting the starter while having someone else try to start the car. Unfortunately, that didn't work at all. I ended up removing the starter, took it apart and cleaned it out, and then bench-tested it with a battery. The starter worked fine. I reinstalled the starter, but still the same story. I did end up getting the car to start by first turning the key to START and then using a "screwdriver" at the starter electrical terminals to pass power from the battery power supply to the starter. Does this mean the solenoid needs replacing or could it possibly be the neutral switch. The key must be turn foward in order for the me to start the car with the screwdriver. Any ideas???

when you turn the key to START position measure the voltage at the ppl wire (terminal S) on the starter solenoid you should get battery voltage, then measure voltage at the B terminal of the starter solenoid it should be constant 12 + Volts. if both check OK replace solenoid.

I can't say for sure about your car, but some cars have a starter relay in the fuse box under the hood. If it does, it could explain your problem. The relay may be sticking. Just a thought. that's the only way I could explain it starting with the key out of the ignition. That most likely wouldn't be a starter solenoid doing this since it doesn't get power until the key is in the crank position. Unless its shorted internally with with B+ voltage that powers the starter motor itself. I guess that's not as likely, but possible. Or it's possible that the ignition switch itself is failing.