Thanks for chiming in Andy and everyone.
I hear it acoustically and amplified, but it's more obvious amplified, at least partly because it's in the decay of the note, which stays audible longer amped up, even clean.
Tailpiece is "Kluson Stop Tailpiece (nickel-plated zinc, steel studs)".
Break angle over the bridge strikes me as moderately sharp, but I haven't had a guitar with TOM-style bridge in a long time to compare. Tailpiece is almost but maybe not quite as low as it'll go. Here a pic if that's helpful.View attachment 111671
I'm somewhat reluctant to raise it, just because in all my years of playing I don't think I've ever done that. Just turn the studs a half turn CCW or so, that's it?
Then it happens on the next higher fret, at the same pitch as before.
That seems to make a fret issue less likely too, and it doesn't sound like any fret sound I've ever heard anyway. It's almost like another string ringing too, but:
a) don't hear it until the note dies out some
b) it stops instantly when I stop the high E, no ring after that
c) I've plucked every stray length of string (between bridge and tailpiece, nut and pegs) and didn't hear that note
d) I used a capo to fret the note so my hands were free, and damped all those stray string lengths, no difference
e) doesn't happen on that same note on the B or G strings
Strings are the ones it came (guitar was used) with. They still seem fine to me; not a pro any more, day job, don't get to play all that much, or work up a serious sweat. I'll be changing them soon anyway, will see if this changes.
Thanks again for any ideas.