Sometimes using a tool post mounted drill chuck is handy for drilling a deep hole as it's easier to use saddle movement to clear the chips. The trick is getting the crossfeed set correctly so the drill bit is exactly aligned with the spindle. Some of the tool post chucks for sale use an alignment pin held by the chuck, but I've made a simple alignment device that uses the tailstock so you don't have to disturb the work in the chuck.
At the top is a Morse Taper 2 (MT2) AXA toolholder. Since the tailstock on the old South Bend lathe is MT2, I will use the drill chuck from the lathe in this toolholder. At the bottom of the picture is an MT2-B16 taper adapter for mounting a drill chuck. I always keep a few of these cheap adapters in stock as a source of pre-made MT2 tapers for various bits of lathe tooling.
Turning the drill chuck side of the adapter to a nominal size, in this case 1/2", via a sleeve in the lathe spindle (MT3 -> MT2). These drill chuck adapters are usually hard but workable with carbide.
Now grabbing the freshly turned end with a 1/2" collect and with a bit of help from a steady rest, a hole is drilled and reamed in the end for a 3/16" dowel pin.
This part is completed with the dowel pin held by red Loctite.
The piece that will be held in a drill chuck in the tailstock is turned to size and reamed for a tight fit for the dowel pin.
All the bits.
And now how to use it.
You need to have the toolpost aligned with the chuck. Probably a square would be good enough, but in this case I used a dial indicator.
The center height of the morse taper toolholder is already set, so you just need to adjust the crossfeed until the dowel pin can fit into the piece held by the drill chuck.
And with the alignment device removed and the drill chuck installed in the toolholder, the drill is now on the spindle line and ready to use.
So, is it worth the hassle? Normally, no, I would just use the tailstock for drilling, but if you had a few long, narrow holes it's worth the setup time. You will also notice it takes a bit more effort for drilling compared to the tailstock since you don't have the advantage of a screw feed.