If you're a tobacco user, I would suggest quitting if you want long-term improvement. If not, then disregard this paragraph!
Personally I can feel a difference between externally-caused cold and the cold that comes from within, which is related to poor blood circulation. Often it's a little bit of both, and that is particularly uncomfortable and makes it harder to warm up. I'm sure a lot of others can feel the difference as well, but I've never thought to ask anyone so I don't know!
When my feet are cold from the inside, I sometimes take a warm bath to warm them from the outside, but I don't stop until I can feel that blood flow has improved as well. That way they'll stay warm for longer.
You can also have an extra blanket, wrapped (comfortably) around just your feet, and then your regular blanket on top.