Unfortunately, many family history searches hit a dead end or two. This is especially common when families moved overseas, or when you are tracing a maternal line somewhere and she got married and changed names. When this happens, don’t despair. You have a few options.
You can do what I tend to do: leave it and go to another branch of the tree if you have other avenues to explore. Sometimes by tracing a sibling, you can surreptitiously find the information you were missing. More than once I have found the married name of a daughter or sister through an obituary, or the occasional cemetery plot location—people do tend to want to be buried near loved ones.
You can look in places people typically don’t: prison records. Military enlistment or pension records. Newspaper articles, especially marriage, death, or birth announcements. Society pages. Archive Finder. Anything might supply a clue. If you have access to personal effects, look in attics and trunks. Check things like bibles. Read old letters and look up any names or addresses you locate. Even if you feel like you are grasping at straws, there is the potential for it to lead you in the right direction.
Try alternate spellings, especially the further back you go. Some things were taken by dictation, and the way I spell Moseley may not necessarily be the way the census taker wrote it down when he visited my U.S. cousins and wrote it down back in 1930, making it seem as if they didn’t exist. Keep an eye out for these idiosyncrasies, especially if your name doesn’t sound the way it is spelled, or if you suspect relatives went through Ellis Island. Or, as terrible as this sounds, if your last name sounds very generic—Smith, or White, or something like that, it is possible that it was changed from a more “ethnic” sounding name during a war or for other reasons.
Some people do DNA testing to give them more of an idea on where to look when nationality might be a question. It can help, but remember that it is only as good as the DNA database the company has (in other words, if you are part Cherokee Indian and they do not have any identified Cherokee Indian on record, then they won’t know that is part of your genetic makeup). So do your homework on the company and don’t just go with the cheapest option or the one with the fastest turnaround time.
If you have the money and the time, travel to the place your trail ends. There might be information locally that you can get access to, whether it be talking to actual living relatives in the location, or maybe a church or library who does not have their town newspapers online accessible. If this isn’t feasible for you, at least call the local branch library. They might be willing to do some of the digging for you.
When you’re really stuck, you can hire a professional genealogist. They usually have specialties, and may have better contacts to access the type of records you will need to fill in those missing pieces.
Good luck, and I hope this helps!