As Chief said, your hands will get used to it, but somew things will make life more enjoyable.
Throw handwarmers (slap and warms) into your chalk bag. Before your fingers start to become wooden, squeeze the slap and warm for a second or two.
If you are cragging or if you are hauling a bag, throw in a warm belay jacket and a thermos of tea or coffee. Starting out warm with circulation to the hands and fingers helps.
If you can put your climbing shoes inside your jacket, the rubber on the sole will warm up a bit. Climbing rubber looses a lot of its smearing ability when its below 36F (I haven't tested to see where its transition temperature but that would be a fun thing to do on my instruments at work).
Find routes that are facing the sun and life will be fairly enjoyable.
No reason that you can't climb quite a bit in the winter. I worked on these routes when it was between 25F and 45F.
You will also probably see Matt on Lifeline (5.13b'ish) in the next Alpinist (assuming the editor goes with the image) as we were working to get the second ascent (has yet to go) and when the photo was taken it was snowing out. Also this image of Matt was taken when it was 34F out
so go out and go after it. Note: In the above photo you can see Matt wearing polypro underwear, pants, polypro top, long sleeve top and a hat. Find things that can be layered loosely.