If you want to transfer, you must be sure there is no any gap between the two J1 program. For example, if the termination date is Aug 20, your new J1 must be effective at Aug 21. Otherwise, you will be out of status and you have to leave the states. After termination, there are 30 days of grace period, and they are for the preparation to leave, but not for transfer. So make sure to get new J1 form before termination. You do not need to tell the supervisor too much. The international office should help you for the transfer.