I work for a search firm, granted in Healthcare, but we deal with people making career decisions like this all the time.
By the time there is the face to face visit - you need to be ready to accept an offer.
The fact that he wasn't, and wants to take a day to think about it, is a MAJOR read flag.
If he was seriously considering - the discussions with family would have already happened.
Anytime someone needs 24 additional hours after a face to face visit - your chances fall to less than 5%.
Don't mean to be negative here, but I wouldn't get your hopes up on this.