The way I see it the sooner you get him in there the better.
It means he adjusts sooner and the stress isn't drawn out
and getting him in a clean, warm, safe environment is the most important thing right now.
I've had 9 bettas and all of them have adjusted really well and fast to a new environment;
as long as you give them some alone time to get comfortable.
The fact that he seems happy means that he is comfortable with you and being able to see you (after give him the night to adjust) will help him.
A betta I gave to a friend a week ago still came up to me today when I randomly showed up.
Spend sometime with him the next day and he'll feel more comfortable.
Betta aren't nearly as delicate as you think; he'll be fine. :) *pinky promise*