You just need to dial it in so the turned section is centered & parallel to the lathe bed. Drilling from the TS will center itself; you might prefer to start the hole from the end without the center hole already drilled. Or, after dialing the piece in, check to make sure the existing center hole is actually still on the axis of rotation.
You didn't mention the hole size, so there may be some other options. The easy one is getting a set of shorter drills. Twist drills come in 3 standard lengths; jobber, screw machine & taper length. Jobber are the commonly found ones, screw machine are shorter, taper length are longer. The sleazier approach is holding a drill in a carriage mounted holder by clamping down on the flutes with a piece of brass tubing to protect the flutes. If you do that, you'll appreciate that its such a nuisance to set up you'll be inspired to get a set of screw machine length drills ;-)
If the hole is big enough, it's also possible to drill it as large as possible with whatever size drill is short enough to fit on the lathe & finish by boring to size.
Roy
