Good idea to post your methods, Ubarch
Performing such a task frequently, makes a good excuse to buy a set of drill blanks (as if we need excuses to buy more tools ). Then, I leave the part sitting unclamped on the tabletop, and lower the rotating drill blank into the hole, then shut the spindle off. Even quite a small drill blank will pull the part into position on the table. So, while the chuck is still lowered, with the drill blank at rest in the hole, apply a top clamp or two, to the part. Then exchange the drill blank out for the desired drill bit.