These constructions both have Google hits: もっと早くに言う and もっと早く言う.
They seem to mean more or less the same thing. But what exactly does this に mean and when can we use に after an adverb?
I don't think we can say stuff like *強くに叩く for just any adverb.
Maybe it adds some temporal nuance, like までに expressing a deadline? Is this phenomenon limited to 早くに?
早く in 早く言う is (functioning as) an adverb (or, the adverbial form of the i-adjective 早い.)
早く in 早くに言う is a noun (the noun form of the i-adjective 早い -- 形容詞「早い」の連用形「早く」が名詞になったもの). So 「早く+に」 is literally "at + an early time".
Similar examples of this noun form:
- 遅く -- こんな遅くに何の用だ
- 遠く -- 遠くで雷が鳴っている
- 近く -- 近くまで来た
- 深く -- 海の深くに沈む
Not all i-adjectives work like this (eg 美しく, 明るく, 大きく can not be a noun.) Only several i-adjectives expressing the degree of time and space have this noun form (eg 遅く, 早く, 近く, 遠く, 深く, 古く, 高く, 低く).
