If you didn't wax the threads or otherwise contaminate it yet.....
Put it back dead centered in the lathe, and squirt a healthy slug of the cheap, thin super glue in, use a wood splint to make sure it goes all the way to the bottom of the hole. The glue should be put in after the cue is in the lathe and aligned, so as not to have the glue setting on one side while that task is completed and indicated.
Put a plastic container on the toolslide that can surround the end of the cue, or put a plastic sheet/apron on the lathe and up the wall, stand clear, and turn the lathe on to a moderate speed. Just let it run for 15 mins or so, it takes a largish slug of cyano a while to harden. However long you let it run, i still would not touch it for an hour. Check with a splint if the glue near the bottom is hard or still gooey.
Verify the shaft is as centered as you expect, take a skim cut to bore it on center, and tap it again using wax. Live tapping is possible, but riskier unless you are dead confident of picking up the original lead.
So far i only use G10 pins, & do this on most of my shafts, just for the extra wear resistance. Might be superstition, might be useful.
smt