"I want a cuter bag."
How would you say "I want more cute bags"? Like, you have one cute bag but you want more?
The sentence as it is has a little ambiguity just because もっと could apply to the whole rest of the sentence or just to かわいい
〔〔もっとかわいい〕カバンが〕欲しいです。- I want (more cute) bags.
もっと〔かわいいカバンが〕欲しいです。- I want more (cute bags).
We can force もっと out of the subclause where it's getting attached to かわいい by moving it after:
But I think probably a comma (or an even subtler pause in speech) would also do: