Steve Walker wrote:

I've never seen a low profile plug...

Annoyingly, I used to have one (and possibly several) branded, of all
things, "BBC" - possibly dating from the time when the Corporation did
actually have all manner of equipment manufactured to its own specs.

...but what's stopping you putting a cutout in one edge of the existing
socket
to run a spur directly from the terminals to the
new one? A bit of a bodge, but very practical.

The main thing stopping me is that it would indeed be a bit of a bodge
(albeit one safely hidden away)...