Invest in a pair of fleece-lined waterproof socks and a balaclava.
If you know you’re going to be in a wet climate, wool socks might not cut it. Used by the U.S. military, these socks keep extremities dry and warm down to temperatures as low as -30º F. They’re $55.95 from Amazon.
A balaclava will keep cold air out much better than a hat and scarf will. Under Armour makes a good one.