2000 Mazda 626, V6, 2.5 Liter, 160,000 clicks on it

My car won't start when it gets a little bit cold out. And not just in -30 weather, but even if it's only -5 or -10. Not sure what to say about the time leading up to this problem. It has always been very reliable starting in the winter in previous years. It has never had a problem starting in warm weather, only recently when it's a bit cold out. If I keep it plugged in all the time it will always start, but if I leave it unplugged over night it won't start in the morning, or really struggles to start. I changed the battery and am having the same problem when starting my car. Any ideas?? Alternator maybe?? Thanks.

I venture a guess here....

One possibility could be a coolant temperature sensor.
Just one of many possibilities..