Notice how the scalene muscles come from the top of the neck and travel down to the front of the first few ribs.  Also notice that the major nerves and vessels that supply the arm go through these muscles.  As they get tight, these structures are compressed, thus causing numbness/tingling, weakness, coldness, and poor circulation in the arm.

One of the key treatments for TOS or Thoracic Outlet is deep manual stretching of the scalene muscles.  This requires a good massage therapist, physical therapist, or Rolfer who can work these muscles from the front of the neck.  You should notice almost immediate relief of hand numbness or some increased symptoms.   You should have a significant decrease in hand/arm symptoms by the sixth visit.   There are a number of other problems that also need to be cleared in the ribs and shoulder (and sometimes the neck) for this problem to resolve completely and not return once the deep tissue work is over.