I have given my winter tips before on other threads, but I'll repeat them again.
The most vulnerable instrument is you and your body, so that is the thing you should look after best. Wear a hat. Oddly enough, you should be careful not to wear too much clothing on the way to the pitch - especially on a very cold day. If you sweat, you can arrive at the pitch soaking and have a layer of cold, wet clothes on for your whole first pitch. This is how you get pneumonia and other lovely chest infections. You should make sure that you have good shoes and a hat. A lot of heat loss is from the head and feet. Until you are warmed up, it is good to have a scarf on a very cold day. It protects your throat if you are singing, and also helps make sure that less cold air is going straight into your lungs. When the temperature is around five Celcius and likely to fall, always cream your hands thoroughly before you go - and keep some hand cream with you. You are likely to get cuts on your fingers otherwise, and apart from stinging like Hell, it can get messy if you start to bleed over your instrument. A little charcoal hand warmer in your pockets, for warming your hands between songs, can be comforting. If you don't have one, simply putting your fingers in your mouth from time to time can help to keep your circulation going. It does matter. Don Partridge lost the use of the little finger on his left hand to frostbite in Canada one year.
The good news is that you can make good money in very cold weather. Then you can buy hot drinks and warm food for yourself and you are not dependent on anyone. Good luck!