Re: Searching Hamburg lists for family groups #records
You didn't mention searching for obituaries using the immigrants' names. It is possible that one of the parents had siblings or other relatives living in the US under their original names. Those relatives might be mentioned in your family's obituaries or your relatives in theirs.
You should also search the US component of JewishGen records. https://www.jewishgen.org/databases/USA/ Your relatives might show up giving a clue. One part of the US search are records from the Seeking Kin database from the Boston Jewish Advocate. https://www.jewishgen.org/databases/USA/BostonSeekingKin.htm For fifty years various agencies tried connecting Europeans with their American relatives by putting ads into multiple Jewish newspapers. Many did not know in what state their relatives lived, so the same ad was published in various states. Some of those ads give both the names used in the "old country" as well as the new name. A search in the US JewishGen records will turn up an ad for a name in any position.
I encourage genealogical societies from other states to contribute ads published in their state's Jewish newspaper(s).