2.Will this person do the job?
3.How will this person to fill with the team?
You only mention your career history one time during the interview which might be a good excuse for the future communication,say you and your new boss will have something in common,but just one thing to let you know,you might make your boss feel very uncomfortable 'cuz you're trying to touch his privacy.he might get insulted and start thinking "Is this person still contacting the old boss? Can he/she focus on the current position? Does he/she wanna go back?" end it up you might burn you ass.
If you really want to bring in more business or create stronger relationship between these 2 companies,do it slowly like no one will notice that,however,ask yourself first to see how much benefit you will get from there.