I've been making a lot of 'cheap pens' - wooden jackets to house ordinary Bic biro infills. They are fun to make, cheap enough to just give away, and make great skew practice. They requires drilling a hole almost 5" deep using a long 5/32" bit that is unfortunately very flexible.
I mount the blank in a scroll chuck in the headstock, making sure that it is centered on the tailstock. I start the hole with a machinist's center drill (using the straight portion only) in a Jacobs chuck in the tailstock, and then put a standard length 5/32" bit in the Jacobs chuck. But to minimize flexing, I put the drill bit as far into the chuck as it will go so that I have the minimum possible extension of the bit. Drill at about 400 r/min, but advance the bit VERY slowly so that the bit cuts its way into the wood rather than being forced in by the advancing tailstock ram. After drilling to that minimum depth, I remount the bit to expose its full length, and continue drilling. Finally, I mount a long (6") 5/32" bit to finish the hole to the full depth. I go in about an inch at a time, back the bit out to clear the swarf, and then drill some more. My experience is that if I'm careful (and don't rush the process), the hole will remain bang on center.