Determiners can be joined with adverbs to describe quantities of countable plural nouns.
Unfortunately, iWeb corpus doesn’t allow us to search for “so many” + collocates. As a result, we can refine our investigation at WebCorp LSE which allows much more wild card searching. It is, however a bit small and uses a different coding system. Looking for “so many” as an adverb and determiner produces almost nothing. As a result, just a search for “so many” with no coding attached shows that they have it coded as an adverb + adjective. Regardless, we can get some useful collocate data without stopwords (words which are the most common top hundred or so) to get data like this:
External Collocates for “so many”
This data basically shows us that the most common word is “people” directly after our phrase, as in “so many people” or on the left the verb “see” as in “see so many“. The tagger on Webcorp has “many as an adjective. And we also know that “many” can be a pronoun. We might then be losing our attempt to find adverbs modifying degree.