The first and easiest place to check is the gear selector linkage rod. The rod has resilient (plastic and/or phenolic) bushings at both ends. If the bushings are damaged or missing, it quite often happens that the lever on the side of the transmission is not moved fully into the D position, thereby holding the trans in 2nd.
If the bushings are intact and the linkage is properly adjusted, then the next step
is:
to check governor pressures. This involves installing a pressure gauge with enough hose that the gauge can be read inside the car while driving, or on a chassis dynamometer.
The third step involves removal of the valve body to inspect for a stuck command valve.

The operating elements, K1 & K2 clutch pistons, and the B1 & B2 band pistons are not on the list of suspects.

It may interest you to know (if you do not already) that your trans, a 722.0xx (3-speed) is actually a 722.1 (4-speed) with 1st gear permanently disabled, hence, it operates in 2nd, 3rd, and 4th.