Extending iphone 13 mini warranty

not sure if they offer apple care for refurbished stuff

but at any rate apple care can only be purchased within a 60 days window after purchase

read more here

Add AppleCare coverage to your Apple device - Apple Support

or reach out to their official support channels

Get Support (apple.com)

And or

Contact - Official Apple Support

And or

Contact - How to Contact Us - Apple

Or

Genius Bar Reservation and Apple Support Options - Apple

How do I purchase more warranty from Apple?

Unfortunately, at this point, you can't. Extended AppleCare coverage would have needed to be purchased within 60 days of the original purchase date of the iPhone. Your insurance company might offer some type of coverage on the phone.

… refurbished Apple 13 mini iPhone …

Apple-refurbished products get the same one year limited warranty as new products get, running out after 12 months.

Additional coverage through AppleCare+ is available only within the first 60 days after purchase from Apple (or Apple Authorized Reseller). This extra coverage is now available for very recent newly released models; i.e. iPhone models 16 (2024) or later, not for the older model iPhone 13 mini.

AppleCare One (for your multiple Apple products) is available in select countries for Apple devices of four years old or newer (headphones must be less than one year old to be added).

With the price of AppleCare+/​AppleCare One, it would probably not be worthwhile compared to the day value of just the older (2021) product.

You can contact Apple Sales (1-800-MY-APPLE in the US) or Apple Support (1-800-275-2273 in the US) for more details. For other countries: Contact Apple for support and service - Apple Support

If you’re in the US you can have your iPhone insured by a separate rider to your home owners or renters insurance. It’s commonly known as a Whole Risk Rider, but to your agent it’s known as an Inland Marine policy or rider. Typical premiums are several dollars per $100 of value. A $1000 phone would typically cost $20 to $30 annually to insure against theft, loss, breakage or any damage that would occur.

Jeff Donald wrote:

If you’re in the US you can have your iPhone insured by a separate rider to your home owners or renters insurance. It’s commonly known as a Whole Risk Rider, but to your agent it’s known as an Inland Marine policy or rider. Typical premiums are several dollars per $100 of value. A $1000 phone would typically cost $20 to $30 annually to insure against theft, loss, breakage or any damage that would occur.

It wouldn't surprise me if there was a deductible as well, in which case it is marginal if it is worth getting any kind of insurance unless you are very careless with your equipment.

Deductible would apply if you don’t list the iPhone as a rider or separate policy. Most companies offer it with a deductible of various amounts. For example if the item is $10,000 value the premium would be $200 to $300 annually. But if you are willing to accept a $500 deductible the annual premium would drop in half. On a less expensive item like a $1000 iPhone the deductible wouldn’t save much premium. It wouldn’t make sense to go that route.

Extended coverage, built in. AppleCare covers battery replacements at no extra charge if the capacity drops below 80%. Plans also include an extended warranty 

AppleCare