Random "POP" when going into gear. 2000 Monte Carlo

I have a 2000 Chevy Monte Carlo SS with 120,XXX miles on it. For the past year, I’ve been hearing a very loud POP or KNOCK when I put the car in reverse from park. Here are the details that make it so vexing.
• It only happens first thing in the morning. For the rest of the day, there is no pop.
• It only happens in cooler weather (less than 50°.)
• The sound comes from different places each time. Sometimes it sounds like it’s in my rear passenger wheel well. Sometimes from the front of the car. Sometimes from directly beneath the driver seat.
• Rarely, there will be times when I’m driving and I “goose” on the gas, there will be a light pop sound. (when in drive)

I’ve had all of the motor mounts replaced (per mechanics opinion) and the problem persisted. My mechanic suggested it might also be transmission mounts but he doesn’t know without actually replacing them. From visual inspection, he didn’t see anything suspicious. Since the car is probably near the end of its life, I didn’t want to spend a lot (more) on something that might not even be the problem. I asked if it was safe to drive and he said yes. I have noticed that when I put my car in reverse each morning and hear the pop, it seems to jar the car rather violently.

Any ideas?

I have a 04 Monte Carlo with about 2/3 the miles you have. The motor mount and trans mounts should be changed at the same time. They go bad relatively early in my opinion. The noise you hear is most likely the trans mount moving around. Ask your mechanic if he used hydrolic mount or rubber since there was an option in the aftermarket world. The rubber does have a more consistent thud when put into gear but I see much less movement when I pull up my driveway and put the car in park, let off the brake and see how much roll back there is. The hydrolic may still let the motor move some if the trans mount is bad.
Also the upper motor mounts go bad but not as often. These are located at the top of the motor to the radiator support. The rubber inserts deteriorate over time and can cause movement and bang.
Next, I have been told the aluminum mount for the driver’s side can also break:
I have not had this issue but it would cause some noise if broken.
One last area, the steering column has an intermediate shaft from the steering shaft to the steering rack under the car. This can get a clearance over time and cause an audible noise in the cabin of the car. You can feel it in the steering wheel in turns and when you change from reverse to drive and some times from park to drive. There is no issue from a hazard because the splines are huge here, just causes a noise. Your mechanic can replace this shaft with an upgraded one from GM if he thinks this is the issue but it is only a nuisance and should not concern you for a safety reason.