Are you familiar with one that you can disassemble?
It sure looks like the guts should unscrew from the cover.
You unscrew the outer "shield" but you then just see the outside of the socket and the wires soldered to it.
I can see now, after a few minutes fiddling with aluminium foil, that the loose connection is not due to the two prongs within the socket which grip the plug (tip and ring), but the neck of the socket where the sleeve touches it (ground). The quality of the lead and jack is pretty good; the others I've look at don't look as repairable. I've tried the two headphones I own with that size jack and they both act the same way and have caused no problems previously on other sockets, so I guess I either return this and spend probably at least double on the next best, or somehow fix this one with some permanent version of "wrapping foil around the sleeve"!
Edit: Yeah, wrapping a strip of foil round the sleeve a few times well away from the ring, then pushing the jack in forms a solid connection. This'll do for now.