I will rephrase.
You CAN get your iPod Touch repaired/replaced without the warranty, but it WILL be a lot more expensive.
Cheaper to get fixed/repaired or replaced under the two-year extended AppleCare+ warranty.
But you should call Apple ( https://www.apple.com/contact/ ) or visit your “local” Apple Store ASAP because if you are right on the edge of being able to purchase AppleCare+, I would check, then purchase and then get your iPod fixed or replaced for a very small fee under AppleCare+
Even if you are over the 60 day deadline by 5 to 7 days, Apple may, at their discretion, allow you to still purchase AppleCare+ for your iPod Touch, then you could have this fixed or replaced under warranty.
Without extended, two-year AppleCare+, the screen repair or replacement is going to be expensive.
The out of warranty costs could be up to 60% of the original cost of the iPod Touch.
Check to see if your iPod Touch is still eligible to purchase two-year AppleCare+ or close or not, then purchase, then get your iPod Touch fix/repair or replaced.
Good Luck to You!