I've had starters stick after sitting eons, the shaft/bendix can get rusty enough to hold them in, also the copper discs in the starter solenoid( if it has that style) can weld up- especially if the thing is cranked a lot with a low battery...low voltage heats stuff up quick...
Kultultz is right though in your case, the problem HAS to be the fender well solenoid, as it shuts off power to the starter mounted solenoid...
The little posts are usually marked R (ignition resistor) and S ( start signal) R is resistor bypass, it feeds full voltage to the ignition coil around the coil ballast resistor, to maintain hotter spark while cranking, usually it's a brown wire. S is the start terminal to the solenoid coil, usually a purple wire. If unmarked, usually the S is the one closer to the battery terminal. You can verify by reading ohms from the mounting bracket to the little posts, the one that's infinity is the R , the one with some resistance will be the S.
If the R/S are reversed, the coil resistor might hold the solenoid on, but the thing might not crank unless jumped by jumping from battery terminal to the S...most old fords can be cranked with a quarter this way, "made for hot wiring"... Was kinda a dumb design as a little jumper was all a car thief needed, jump s to start, quickly move jumper to R to back feed coil and go...I've started mine that way many a time while messing around under the hood...
Just looked at Kultultz's pic above, never saw one with I instead of R- at least don't remember, but my memory is shot... Never knew either that these had been around this long- the B6...part number means 1956? Wow...That's a long time for ANY part to go unchanged, think my sons 1990 Lincoln still has the same thing too...