Do I need to wait until my AppleCare expires to renew?

From the above linked Apple Support document: Extend your AppleCare+ coverage - Apple Support

❝Purchase the new coverage within 30 days after your original coverage ends. To find out if you can purchase the new coverage, go to mysupport.apple.com and follow the onscreen instructions. The new coverage automatically renews until canceled.

Your plan coverage may continue beyond 24 or 36 months, on a monthly or annual basis, until canceled.❞

If your device is eligible, Apple will send you an email invitation to renew. They have done so twice in my case for two different devices. Just follow the prompts.