A friend of mine had a Studebaker Hawk with an overdrive knob on the dash that had to be pulled I believe. I have no idea how it worked. My only idea as to how it worked was that it was in the rear end and not the transmission itself. But that’s always just been a guess
I do remember that the overdrive was in the rear of the transmission but it was not an addition in that everything was enclosed in the same case.
When I had to replace the transmission, I got another one from a junkyard. I had no idea how they worked but remember Daddy hooking up some wires to the replacement and checking to see if the solenoid worked. Sure enough, every time he would put power to the terminals, the solenoid would click and you could tell it was moving something inside the transmission.
The car companies sourced overdrive units from several companies in the '50s and '60s - some were completely manual, some used electric solenoids, some a combination. Some were attached to the tailshaft housing of the transmission (I think the '55 Ford used a Borg-Warner unit mounted there); some overdrives were part of the rear axle assembly (later 1960s Fords and possibly the Studebaker). One such product was the Hone-O-Drive.
Whenever I find a junkyard with some old pickup trucks, I peek under all the '60s and early '70s Fords - those seem to be the last place that those axle-mounted units were used. They are worth $$$$ to the hot-rodding community, the now rare examples bringing ten times their original price of $300 - $400. Nowadays, though, not many junkyard workers know what those things are. I'd buy the entire axle unit, if necessary, to get the overdrive.