Car Won't Start on Cold Mornings

I have a 2005 Toyota Corolla S with 120k miles on it. I recently moved from Texas up north and am finding that the car is not starting in the morning (particularly on cold mornings). The car starts fine later. I had the battery tested and replaced but the problem persists. I tried holding the key in the start position and taping on the starter and the starter makes a clicking noise (like it’s trying to start). I’ve had others tell me that the starter is probably bad, but I’m trying to figure out if it could be anything else or if there are any other tests that I can do myself before I start blindly replacing parts. Thanks!

“Car won’t start” is rather vague. We need more information. When you turn the key, what, exactly, happens?

Does the starter engage and spin the engine, or do you just get the “clicking” sounds, or does nothing happen?

Does the starter only click when you tap on it, or does it click when you turn the key?

Have you checked ALL battery cable connections, including ground?

When I turn the key, they engine does not turn at all and the starter does not engage. However, all of the indicator lights (including the headlights) come on. I have checked all of the battery cable connections and they are fine. I would like to reiterate that this only happens when the car is cold. The car starts fine in the afternoon and in the morning if it hasn’t been cold outside during the night (say in the 30’s).

My money is on the starter (specifically the solenoid)