Temperature outside could be a factor as you mention of course. If you can source a good used relay from a salvage yard or elsewhere is one option.

I am hesitant to mention this but good used relays are going to become scarce to find at some point. So if the relays internal points are still in good shape the day may come where we rebuild/repair the electronic/electrical portions ourselves.

To make sure it is the relay and not some external connection you want to hook up a meter between the relay trigger wire from the ignition and ground. This to make sure youi are getting 12v every time you turn the key to that point. This should be checked at the relay. Also make sure the ground for the relay is good. It sounds like temperature is a factor here so a cold solder joint or connection for example could reflect that.