You can have good luck using a hole saw with noe center drill if you pre drill a hole with 'that hole saw' and clamp on the top of the new part you want to drill. The top guide wood should be 1/2 - 3/4 thick and clamped firmly onto the new piece to cut. A drill press is important for this type of work.
using a hole saw with no center drill is hospital visit waiting to happen!
Bill