Yes! You can start or join a Premium for Family plan right away, provided that your subscription was set up through Spotify directly.
If you'd like to join a Premium for Family plan as a member, you can ask the plan's owner to send you an invitation first. Once you join their plan, future payments will switch to the plan owner and any Premium time remaining from your prepaid subscription will resume when you leave the family plan.
Further, if you want to start your own Premium for Family plan and invite your family members, you also can! Keep in mind though that your next payment date will likely be earlier than it had been, due to the difference between the price of Premium for Family and regular Premium. Just head here to get started.