Dave, I wonder if the 1962 change didn't apply to those aircraft because of the use to which they were put, VIP transportation, particularly President and Vice President etc. All went to the 89th Military Airlift Wing which took over from the 1254th ATW in 1966. Logic to me to have the aircraft assigned the same designation for maintenance purposes if nothing else, since except for the two EC, all aircraft were operated by the one unit supplying VIP transportation. There is always an outlier in these things, or "what about". The designation covered nine aircraft
VC-137A 58-6970, 58-6971, 58-6972, re-deseginated VC-137B after re-engining, C-137B following removal from VIP role
VC-137C 62-6000, 72-7000 Presidential aircraft, C-137C when Presidential role removed
C-137C 85-6973, 85-6974 second hand, one ex Buffalo Airways, one that had been seized buy US Customs, one airframe later becoming an E-8C
EC-137D two airframes built as Early Warning and Control System prototypes. Later re-engined and re-designated E-3A.