I love love love my tall winter boots from Ariat. I have 2 pairs, one given to me free becasue the lady thought they were worn out, one that I got for less than half price, barely used. Look on e-bay, or at used tack stores, but find a pair of winter tall boots, they make a huge difference.
I wear soccer socks or ski socks under the tall boots.
I warm the bit up in the sink with warm water rather than using my hands because if I use my hands then my hands are freezing. Friends fill a sock with rice and pop it in the microwave before they leave for the barn, and then pack that around the bit to warm it up.
You could also do that and put a pair of those in your pockets to keep hands warm whil tacking up.