My overdrive button won't work when car is cold.

Will the car go into overdrive under normal driving cold or warm? Or are you just saying if you push the button it won't drop into 3rd? Does the light come on in the dash saying o/d off? Let me know..see ya.

Since the button should press down a little bit, it sounds like it the switch simply took a poo poo, and in cold weather, the temps affect the broken switch in a mechanical fashion.

FWIW, AFAIK, it is just a momentary switch. You should be able to use a momentary switch from the parts store for a little bit till you get the correct new switch.

I am pretty sure that when guys add another switch (to put it on the shifter handle, etc) that is all they do.

If you use the OD button a lot, have you seen the shifters with the holder for the OD button? The button is at the shifter handle, rather than the console. Seems kinda cool to me. UPR sells a version.
And while installing a new switch, that is a good time to do the retrofit.
Good luck.

Some of the info here might help.

And UPR has a shifter that comes with the bracket they use (so you dont have to fab one up).

Good luck.

Mine stopped working one time after I spilled some coffee or something on the switch. I poured some water on it, pushed the button a bunch of times and wiped it with a paper towel to get the sticky stuff out of the switch. It started to work after that. Your switch could be gone but it might be worth a try.