Can coverage past the 3-year AppleCare plan be purchased for a MacBook Air?

This document describes how to extend coverage:

Extend your AppleCare+ coverage - Apple Support

It states, "Purchase the new coverage within 30 days after your original coverage ends."

It depends on whether Apple offers a monthly or annual plan in your country; if so, those can last for as long as they're able to service your model.

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You may be given the option of changing to that when the three years expires, but not right when you've started it.

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