Here's the definition with examples of like as an adjective that I just found in the LDOCE Online, my friend:
similar in some way: The second dispute was sorted out in a like manner. They get on well together because they are of like mind. Try to buy two fish of like size.
We can clearly see that like is an adjective in these sentences. I don't think we can say the same for like in the sentence under discussion.
I have a hunch that more like is a pat phrase that's used in certain expressions, this sentence containing one of them. I imagine that if we agree that more is an adverb in this usage, then I have to assume it modifies grow.
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