Usually when I want to countersink a larger hole in a smaller hole I use a Forstner bit to drill the larger hole first. The Forstner tip leaves a nice little center hole which enables me to accurately drill the smaller hole all the way through. Sometimes I get overeager and forget to drill the larger countersink hole first. One is then faced with the problem of centering the Forstner bit on a hole. Not so easy!
This happened again the other day and thinking about the problem for a while, I came up with the following solution. Put a dowel in the hole after marking the center on the dowel. After drilling the countersink hole, just remove the dowel.
-- Integrity is doing the right thing when no one is watching.