I don't think lathe tailstock tapers typically have a slot for a drill tang. They rely only on the taper to hold.
You can probably improve things by cleaning up the taper with a #2 Morse finishing reamer. You don't have to take off much, just enough to knock off the dings and nicks. I clean up the taper on my lathe about once every 5 or 10 years, it seems.
Try to make a living, not a killing. -- Utah Phillips
Don't believe everything you know. -- Bumper sticker
Everybody is ignorant, only on different subjects. -- Will Rogers
Law of Logical Argument - Anything is possible if you don't know what you are talking about.