Offer arrived, can I still ask for more?, data-entry clerk here

It is the first offer in 6 months and I do not want to mess it up. I have read a lot of threads here and could not find my exact situation. I have a competing interview next week. What wording worked?

I thanked them and said I was excited but hoped we could talk about the pay. They came back with a bit more.

Thanks for writing this out. I went through something close to it 6 months ago, and it settled down once I kept a simple log of every call and form.