Yes, I think so. "Daha" is an adverb, so you can use it to intensify many things, but not a noun. If you want to use "daha" with a noun, you need a "çok" or "fazla" between them. So you could potentially say: "daha çok/fazla bir kere" -- but I'm not sure quite what that would mean! Maybe "more one time(s)" -- which does not sound right. If it means anything, it seems like I am asking if you could increase the frequency of doing this thing one-time-at-a-time? But that doesn't make much sense, so let's not go with that.
In this sentence, adverb really belongs modifying the verb:
söyleyebilir misiniz? > can you say it?
[bir] daha söyleyebilir misiniz? > can you say it [again]/more?
And then you are applying a quantity to that:
Bir kere [bir] daha söyleyebilir misiniz? > can you say it [again]/more one time?
--Danika_Dakika