Asking for additional rent can not be done "after fact".
I usually give tenant a few days up to a week for free for them to settle down if the move-date does not match well.
You would have the house sit there anyway.
(I don't like people who can move-in right away - that shows the family has no planning and a strong sign for future trouble).
If you do it ahead of time, you can negotiate, but I wouldn't.
Again, landlording is for the longer-term. You don't have to make money everyday.