Check the records of immigrant organizations in the US.
Scandinavian immigrants typically belonged to a village hall or other organization back in the old country, and they brought this tradition to America, too. These new social clubs included the Sons of Norway, the Danish Brotherhood, and the Swedish Vasa Order. In the peak immigration years, these groups provided unemployment benefits to members, helped struggling families, and sponsored community centers.
The Sons of Norway remains a vibrant nationwide organization, with a magazine, financial products, and cultural offerings. The Vasa Order also continues, and has an archive that might benefit your research.
Although the Danish Brotherhood in America no longer functions as a nationwide organization, its surviving membership rolls are one of the most important sources for clues to family origins in Denmark and to original name spellings. A death index of members is online; the original records are held at the Danish American Archives and Library in Blair, Nebraska.