From the posts I've read, that should be a mechanically activated odometer. If that's correct, the odometer and trip odometer 1/10 mile digit wheels are driven off of a worm gear on the speedometer cable. If the rest of the digits are changing with the 1/10 wheel, then something is binding. Not sure how easy this is to remove and repair. Back in the '60's it wasn't too bad. Since the advent of anti-tamper legislation, it has gotten more difficult to take apart and service odometers. Worst come to worst, you might have to buy a used speedometer and install it. If you do dis-assemble it for inspection, do so on a clean surface. The speedometer portion has a disk magnet that rotates inside an aluminum cup. The clearances are tight and any crud that gets pulled onto the magnet will only compound your problems. It is a somewhat fragile assembly.
Am I showing my age with talk of 'back in the '60s? I'm 53 and have a 3 year old grandchild. :-)