Once you create a new period, you can transfer contacts individually from their personal page. This is helpful when there are individuals with whom you want to continue following up with during a new period. 

To transfer their information, open the past inactive period from which you will be transferring the contact, then go to the contact's individual page. You will see the following button:

When you migrate this contact to the new period, they will keep their contact info, but they will have a clean follow-up slate. The contact is now a linked contact.