Turns out that it was a client from TOJ!
He started out with: "Hello. You don't know me...." He got my name from a colleague of his, whose name I vaguely remembered. Apparently they had tracked me down to TNJ, and called me.
About a year ago, I (helped) build a site with a custom add-on module for them. Now they had a proposal (from TOJ) to use that module/those techniques/that functionality on another site, and they thought the price was too high.
Could they hire me to port the module over to the new site? Would I be interested in attending a meeting, say, for an hour, in exchange for a nice bottle of wine?
No on both counts, for three reasons. One: TNJ isn't in the web development business. Two: I'm not going to do any work in my spare time. Three: Whatever the reasons that I quit TOJ, I am not going to undermine their business.
(Also, the fact that this particular site was the last one I worked on at TOJ, and it was the last 'list of 75 things that need to be fixed before the site goes live', I have no intention of ever touching it again.)
After that rather weird call, I immediately called TNJ. Turns out that a delegation was en-route to that particular client, to talk about it. At least now they know what is going on there.
I hope I don't get any further calls of that sort.