They are a long way down the blood distribution system. They are whizzing around in the breeze, making a really thin boundary layer of cold air sucking out heat. Wearing extra layers of socks usually restricts blood flow to the feet, making matters worse. But far better, take a pair of polartech socks or neoprene socks over polypropylene liner socks with good light-weight shoes will give you the protection you need. Bean's Snow Sneakers are light weight and nicely insulated, they will do the trick.
Regular sneaker are fine too.
As long you have the protection mentioned above. Some people recommend Saran Wrap over the sneaks. Make a chart of temps and the clothing that works best for you. It helps prevent over-dressing. Below is an example chart.