I'm 86.7% sure that what I say is correct.
The 1960-62 frames were wider in the front (at least the front horns are wider) than 1963-66. I'm pretty sure the cab mounts are wider apart too.
The catalog I have shows different part numbers for the 1960-62 front bumpers AND rear Fleetside bumpers.
Rear stepside bumpers are apparently the same 1960-66 (same part number).

I own an original 62 rear stepside sport bumper and it's identical to the 1963-66 step bumpers but the mounting brackets are very different (also verified by my catalog).
The interesting thing is that a 1967-72 (maybe up to 1987 years work too) front crossmember and suspension will swap under a 62 frame just fine. Lots of people do this when they delete the torsion bar suspension.