The Mitchell is for cars with a torque tube, it won't fit any 49 and later Ford passenger cars.
52, 53, and I think 54 are all the same side loaded transmission.
55, 56, and maybe 57 are the same top loaded transmission. Somewhere around 57 the input shaft switched from course to fine spline. The trick is to measure the total transmission length and distance from the front of the transmission to the rear motor mount. Get an overdrive from a Ford with the same measurements and it will bolt into your car. Check for fine spline vs coarse spline at the input and output shafts. You may need to switch clutch disk and/or drive shaft yoke.
I recently bolted, with no modifications, a 56 overdrive transmission into a 52 sedan. Make sure to get a working solenoid in whichever voltage, 6 or 12, your car is.
I have installed 5 Borg-Warner overdrives. I never bother will all the factory controls. You can install them if you find them and you want them, but you don't need most of them. All that you have to have is the lock out cable, a good working solenoid, an on-off switch capable of a few amps, and two wires. I take the governor off, remove the gear on the bottom, and put them back in to fill the hole. Then I run a wire from the "on" post on the ignition switch to my on-off switch, and another wire from the other side of the on-off switch to the solenoid. The solenoid has two wire connections, only one of these is needed. Put power to each one separately while grounding the housing and you will find out which one is necessary.
To get into overdrive with the switch set up you: push the lock out cable in, drive up to the speed where you want overdrive (typically 40-50 m.p.h.), turn the switch on, let off the gas, and you are in overdrive. To drop out of overdrive, turn off the switch and let off the gas a little. You can go into and out of overdrive at almost any speed this way.
With either full factory control or switch setup, never pull out on the cable unless you are accelerating while out of overdrive, or you are at a dead stop.
If this is all a bit confusing, PM me and I'll try to clarify.