Yes! You can start or join a Premium Family or Premium Duo plan right away, provided that your subscription was set up through Spotify directly.
If you'd like to join a 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. Any remaining Premium time you have is forfeited.
Further, if you want to start your own Premium Family or Duo plan and invite members, you also can! Keep in mind though that your next payment date will likely be earlier than it had been, due to the price difference between the plans.