For searches, Austria and Hungary records are co-mingled name wise - the Austro Hungarian Empire. Also check records for people in Bohemia, also in Empire and co-mingled.
And if the American name is like Joseph, check for Yiddish variations like Josef , Yusef, David check Dawid, Moses check Moishe, Murray or Morey check Moritz , Benjamin might be Baruch , Abraham check Avram & Ibrahaim etc
I learned working in the Holocaust Project Jewish names aren't always what you think they are LOL !