Biugh
Bought some small chucks with 6mm hex shafts- ideal for drilling small holes with the piece held in the lathe chuck!Drill Chucks
Handle
Rather than hold a drill bit with pliers or mole grips & risk scoring the shank, you could just hold the bit in a small hand held chuck. As long as you make a central indent larger than the bit, it will self centre. Same applies when using a spindle gouge to bore a central hole. None of this wobbling all over the place & making an oversize hole.
I added an old handle that was not likely to get used for anything else.
Most Jacobs type chucks, whether hand or key tightening are threaded 1/2"x 20 UNF for 13mm capacity chucks & 3/8"x24 UNF for 10mm capacity chucks. The threads are mainly female but there are a few with a male stud. Source a suitable bolt, cut off the head & locktite the thread into the chuck. Drill out the handle to suit & epoxy the other end of the thread into the handle. I drilled all the way through & added a pin to prevent the stud from turning.
View attachment 82795
This drill chuck is mounted onto a 3/4x16 UNF adaptor, which was made for me by a club member, to fit my first lathe. I could use a another adaptor to mount it but it's just as quick & easy to hold it in the chuck.
View attachment 82796
View attachment 82797