I finally found what that extra wall-wort was for.
On my old computer, which has finally been exiled to the basement, I had bought a small external USB hub --- not much wider than a USB connector, in fact. I attached it to the side of the system unit with the sticky tape that came with it ... which meant, of course, that it kept falling off. And, of course, the hub was powered.
So, during some cleanup operation, the hub fell down a crack, the wall wort got unplugged from it ... and the rest is (trivial) history.
I hate to admit this ... but it's a relief to know what it was. I hate unanswered questions like that ...