John K Jordan
If you are drilling first then turning to size later I suspect the drilling is not the problem but the holding. You said you make cone-shaped jam chucks. How are they held on the lathe?
I use jam chucks too but instead of cones the pieces I jam on are cylindrical, more like short mandrels. (I generally don't like cone-shaped supports.) For the headstock end you can wood in a chuck and turn a short cylinder that fits snugly in the hole to hold the head end. (The mandrel will need a slightly larger shoulder at the end to keep if from being pushed too far into the hole in the work.)
For the tailstock end drill a hole in a short block and use a 3/4"x10tpi bottom tap to cut threads to fit the live center. Thread the block onto the center then put the live center and block in the headstock spindle (locking it from turning with a piece of wire) and use it as a drive center to turn a second short, snug cylinder to hold the tail end.
Before I had a tap the right size, I would simply drill a hole in a block sized to twist/jam onto the live center threads then turn a cylinder on it as before.
The way just described is actually the way I USED to make short concentric wooden mandrels, works well but what I do now is quicker: turn a short morse taper on the end of a block, reverse and jam it into the headstock, then turn a cylinder (holding it with the tailstock while turning). I use this method to hold everything from thin spindles to various other jam pieces. The morse taper doesn't need to be very long. Here are some on the ends of thin spindles.
I make a little gauge to aid in sizing the taper but a couple of calipers work as well. Cut a shallow relief in the center of the taper to it contacts the inside of the headstock taper in just two places.
Unlike holding the support piece in a chuck, these can be remounted in the headstock in perfect registration every time.
I don't have a picture, but I often do the same thing for the tailstock end - quicker than drilling and tapping. A short #2 morse taper fits perfectly in the #2 taper built into a Nova live center. I have a bunch of these in the drawer, different sizes, ready to turn to the exact diameter needed every time I need to hold the tail end of something that has a hole.
The Nova is the most versatile live center I've ever had; it is so useful I bought two.
It's cheap now too - I paid $90 for the last one. (Maybe I'll get another...)
BTW, when drilling a deep hole with a Forstner bit I started doing something a decade ago that helps a lot: direct a continuous stream of compressed air into the hole behind the bit while drilling. It not only clears the chips but keeps the bit a lot cooler. No extra charge.