Yes, both sentences are perfectly correct and very common when ordering food or drinks. You can use them interchangeably, but here is the tiny difference in how they 'feel' to a native speaker: 'Can I have one more of the same, please?' This is slightly more specific. It emphasizes the quantity (exactly one). It is very polite and common in formal or semi-formal settings. 'Can I have another one of the same, please?' This sounds a bit more natural and casual in everyday conversation. 'Another' implies you want an additional item of the same type you already have. For Native Sounding English: If you want to sound even more natural at a restaurant or bar, you can also use these shorter versions: "I'll have the same again, please." "Another one, please." "One more, please.