My Shure 55 'Elvis Mic' hás a problem!

The set screw in the plastic connector body has backed down from the connector housing in the bottom of the mic. Turn the set screw so that its fully recessed into the plastic connector body. Connect a female XLR to the male pins of the connector now presumably hanging by tiny wires and line up the bump on the rim of the plastic body with the groove in the metal mic housing. Slide the mated XLR connectors into the mic body until the set screw lines up with the hole in the body and turn the set screw to lock it into place. A blob of silicon adhesive in the set screw hole 'might' prevent this from happening again.