Apologies if this is a dumb question:

Sometimes I have to remount a stretcher on the lathe after I've trimmed, say 5/8" tenons on both ends.

I have pin jaws on my chuck so I'm good at the headstock. But re-marking the center on the live end doesn't work well. I could mount a second chuck on the tail stock, but that's expensive. The tenon is too large to grab in a drill chuck.

This got me thinking, if there was a live center with a convex cone instead of a point, it would self center the tenon, and would be quick. Does such a thing exist? (I suppose I could make one with a 1/2" tenon that I could then grab in the drill chuck...

Alternate suggestions welcome.