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 and you will see the following button:

When you migrate this contact to the new period, they will keep their basic information as well as any custom fields you have created, but they will have a clean follow-up slate.