Hellow. There is no problem with Korean grammar I think. (I'm not a teacher but a native Korean) However, I'd like to say the first own is the better. Let me check the sentences again.

-그는 또 늦게 왔어요.: Good to go.

-그는 또다시 늦게 왔어요. It's good, but I feel a bit of awkwardness. It is depending on the sentence. For example: 그는 어제 취해서 늦게 왔는데, 오늘 또다시 늦게 왔어요.-> It is not weird. Because it has a privious sentence that supports the sentence. Did you get the difference?