Sorry, I wasn’t clear. The problem is the screw socket in the searchlight head. In my version, the plunger contacts a spring tab which is insulated from the searchlight head. This tab goes into the searchlight body and, presumably, is soldered to the center contact of the socket. I wanted to fix it properly but settled for a workaround. I cleaned the socket and added a dab of conductive grease. Then I added a little more solder to the base of the bulb. A touch of conductive grease on the spring tab and both ends of the plunger and it works.
I think in the older version that knob at the back is the key. Looking at it you can see the spring tab and insulator are both under the knob. It was, probably, pressed in at the factory. I guess they didn’t think that someone would need to fix it 74 years later.