If the tail light and the turn signal work, then the issue is not at the socket. A bad ground would cause all three to fail. There are two wires to each tail lamp. One feeds the tail light itself and the other handles the turn signal and brake light half of the bulb. It's the wire with yellow thread in it's casing. Hopefully, there's enough good wire to see the difference.
Here's a link to the 1953 wiring diagram. It is the same as your 54 except that it doesn't show the pigtail for the upper lamp.
One thing to try, remove the bulb from the upper lamp and see if your brake light now works. I've seen where the wrong bulb will cause the lower to act erratically.
https://circlekf.com/kflit/SM/kfsm700/kfsm700028.jpg