Just be sure to check that the tail stock axis is in line with the headstock. And that's not a bad thing to check and adjust anyway. Makes for tail stock drilling that's a lot more accurate and less surprises of all sorts.
It's easy to do. Stick a stub of something into the chuck and turn it to a size near to 1/2 or 3/8 diameter. It's not really that critical as you'll seen in a second.
Now using your handy dandy tool post mounted dial gauge.... er..... you DO have a TPMDG don't you? If not see pictures below for one VERY handy lathe gizmo. The tip is set so it aligns with the center of the head stock axis by eye. Just zero it to the little stub or center mark left by a facing cut on some scrap or your workpiece by eye and it's good enough.
ANYWAY... to check your tail post for on axis you chuck up a stub of round stock and turn it so you're not including any jaw runout. Turn it to a size which you can match with a stub in your drill chuck in the tail stock. A piece of 1/2 or 3/8 O-1 drill rod works well. Or you can use the parallel portion of a fixed center and turn the scrap to a size close to that size. Measure and take note of any size difference between the stub turned in the chuck and the stub or center in the tail stock. Now run your dial guage in and pick up a measure off the stub in the chuck. Move along the bed to the stub or center in the tail stock and note the difference. Now with a touch of math and knowing the difference in size between the two measuring points it's easy to see if your tail stock is aligned or not.
This is mostly for lathes that have a tail stock which can be offset to use for turning tapers. If the small machines don't have this then you're mostly testing for any error that you'll then need to take into account when doing anything fussy and demanding for accuracy. It could also result in the handy threading tool not cutting all that axially.... which was why we checked for this in the first place to go along with this thread..... I got wandering again, didn't I.....
Anyhow, here's a couple of pictures of the dial guage mount that fits into my four position tool post. Use the idea to suit your own machine. If you use a lantern style you may want to add a foot screw so it can be adjusted and reset's to the same riding position each time in the lantern and saddle key. This one is pretty crude. I made it in a hurry. But it's worked so well that I've never bothered to make up anything more fancy. It's just there for all the times I need it. And it gets used a lot. Far more than my magnetic base. And for zeroing work in the independent four jaw this trick really is the way to go.