Maybe someone converted to a solenoid started and installed a push button in a hole they drilled into the dash? That location is where the solenoid-starter push button is located on Hydra-Matic-equipped 53-55 New Design and 54-55 Advance-Design trucks (and, maybe on Forward Control trucks/vans from 1948 and newer?).

Are there still wires connected to the rear of the push button?