It's one of those jobs that may go better using a hand drill (i.e. you
spin the handle round, and the bit goes round).

Have an assistant watch and direct you to stay parallel.

Reason for a hand drill is things go wrong more slowly and your
assistant has time to correct you before you're "in too deep".

You might be able to use the hole in the original cut off bit as a
drilling jig instead (say cut off a 1" slice to act as your guide) -
giving you both the centre and direction.

Remember also, for a good fit, that saw cut must be square - chop saw,
mitre box or a fair bit of handsawing skill.