Applecare for a product not purchased directly from Apple

No AppleCare plan is longer than 2 years. And that means 2 years beyond the initial, standard 1 year warranty for a total of 3 years.

And that 3 years is always from the date the device was purchased. Not when you purchase AppleCare. When you can buy it always varies. You have much less time to get it for an iPhone than an iMac. You can check here to see your purchase options:

AppleCare Products - Apple

There is no way to extend the allowed coverage. Not even directly from Apple.

Actually, that no longer is the case. Apple Care can now be extended past the initial 2 or 3 years.

It can be extended past the 2 year or 3 year coverage as explained here:About continuing your AppleCare+ coverage for iPhone, iPad, and Apple Watch - Apple Support

If you paid upfront for 24 or 36 months of coverage, you can renew coverage on a monthly basis after the 24 or 36 months is over.

Does coverage continue beyond 24 or 36 months if I signed up to pay monthly?

Yes, if you signed up to pay monthly, plan coverage may continue beyond 24 months, on a monthly basis, until canceled.

Well, yes, you need to keep paying monthly for it, and the caveat is, Apple can choose to unilaterally end it at any point arguining loss of availability of service parts.

https://www.apple.com/legal/sales-support/applecare/applecareplus/docs/applecareplusnaen.html

Apple may cancel this Plan for fraud or material misrepresentation, or if service parts for the Covered Equipment are not available, upon thirty (30) days’ prior written notice.

As long as its within the 60 day period to buy Apple Care, you should be able to.

See here: Buy an AppleCare plan - Apple Support

Ah! Thanks for the info, Phil. First I've heard of it.

Bet it gets more expensive the further out you go.