The desire to know where an immigrant ancestor came from can be a strong one. The problem is that sometimes finding enough information to make that determination is difficult, especially if the immigration took place before 1900.
The best approach is to research the immigrant in the country of settlement in an “extremely exhaustive” manner. It’s difficult to define that phrase, but essentially it means locate everything that may directly or indirectly mention that person. Those records may mention where the person is from (either specifically or generally), they may suggest associates of your ancestor who knew them “back across the pond,” or provide additional clues.
Generally speaking it requires that the researcher learn about all records that exist where the immigrant settled including civil records (local, county, state, and federal) and private records (church, funeral home, newspaper, occupational, etc.). It requires searching those records even if you think your ancestor will not be mentioned in them or that they “probably would not help.”