I wash my horse in stages and leave her rug on the places i'm not washing. Ok, here's what I do.
First, I undo the straps on her rug and give her a haynet. I fold the rug back and the back half of her. Then I dry this area off as much as possible, put a dry towel over it and put the rug back down. Next, I fold the rug away from her front area and wash this, dry it off as much as possible, put a towel there and do up the rug so its on properly. Next, I put towels around the base of her neck so water doesn't drip under her rug and I wash her neck and mane and dry them afterwards. If she's very cold after this, I leave her legs until a few hours time, Otherwise I do them last.
Normally, I put on a fresh rug once she's dried a little because the rug tends to get a bit damp from this washing process.
Another thing that I've used when its REALLY cold is using warm water from the house. Dad wasn't very happy when he didn't have any hot water for his shower that night though hehe
Anyways, give it a try and you might find it helps