[pminmo]

If I put a 1/4" bit in my tailstock chuck and advance the spindle a
few inches, I get significant play, probably around .05in. Is there
an adjustment or modification that I can improve this?

Phil

[HuFlungDung]

I'd be careful and take a look at how far out the spindle moves out before it comes right off the thread. You don't really want it hanging out that far, and by the sound of your description, you may be doing just that.

Use a stubby spotting drill or a center drill to start the hole in the end of the workpiece. Don't expect to start a flimsy drillbit in the end of a piece without an accurately placed start mark.

The end of the part also needs to be faced square before you attempt to drill into the end of it, or it will force the drill to wander off. You might be surprised that even on a large lathe, relatively small details like this will flex a 3" diameter spindle.

Another thing to look at is the accuracy with which you are chucking the part. If the start dimple is off center because your chuck is no good, then the drill will wobble as it attempts to follow that dimple, and flexure of the tailstock will ensue.

[Konrad]

It could be worn out??
If that's the case, the spindle can be chrome plated and ground oversize to suit the honed I.D
Konrad

[JFettig]

The tailstock on the mini lathes kinda suck in a way. make sure you do as huflungdung said and check the clamping of the tailstock. It is very easy to clamp it down and have it be off center.
Clean off the ways and the bottom fo the tailstock to make sure.

-edit-
wow its early I now see what your saying, yeah, its really not all that high quality stuff make sure to start your hole with the quill, or spindle or whatever as far in as possible, that way it has something to guide it. yes I agree, its kinda wack. That is one of the things I want to do with mine, make a new tailstock.

Jon