Can I still get Rice if I have Fine Snow from a previous playthrough?

The Divine Child gives Rice to the player occasionally (I’m not exactly sure what the triggers are), until you give her the Serpent Viscera, at which point she gives you Fine Snow.

Upon starting NG+, I still had a Fine Snow in my inventory. Upon reaching the Divine Child again, she gave me the Rice, but I am yet to receive another.

The wiki states (about the Rice item):

The Divine Child won’t give this item to Wolf, if he already has a copy in his inventory.

It doesn’t specifically state anything about a cross-over between the different items, however.

So what do I need to do (and not do) in order to get another Rice item?