Hello. In my opinion, it does seem as if "tad" functions as an adverb, although an article precedes it. I found an insightful explanation of the word/expression in a book titled Common Errors in English Usage by Paul Brians.
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vocabfishSome people combine this with the equivalent expression "a bit" and say "a tad bit". This is redundant. Just say "a bit" or "a tad".I agree with Mr. Brians. It sounds completely redundant to me -- though I admit people do say it.