Is it possible to improve your Cave of Shadows saved data after unlocking the final section?

The game only saves data to the Amiibo if you've completed the cave, or as much of the cave as you've unlocked, with more hearts remaining than your previous time.

Your Amiibo currently has data with 13 hearts remaining, so you'll only be able to save again if you finish the entire thing with 13.25 or more hearts. If you finish it with all 20 hearts, you won't be able to save into the Amiibo any more.

It's not clear whether this is intended or a bug. On one hand the main use for the data in the Amiibo is to restore your hearts (we don't yet know how Zelda U/NX will use it), so you'd want the most of them you can have. On the other hand it also saves other data (such as which other Amiibos you used) so perhaps it should give you a chance to decide you'd rather save fewer hearts but a better overall "score".

Update as of 2017-01-31:

We now know what the "Zelda U/NX" (titled Breath of the Wild) will use the amiibo for: To conjure a wolf to help you, which has as many hearts as are saved in the amiibo. So it makes sense for it to only remember the time you finished with most hearts.