Transferring Apple Watch from a parent's phone (currently using Family Setup) to a child's phone. Do I need to do anything different if the watch is currently paired with my phone via Family Setup?

My son's Apple watch was setup on my phone using Family Setup. He now has his own phone and I'd like to pair the watch with it. I found the documentation for transferring a watch to a new phone, but it doesn't mention anything about Family Setup. Are there any extra steps I should be taking if the phone is currently setup using Family Setup? He doesn't want to lose his contacts and I'm trying to avoid the need to manually reenter the contacts on his phone.