Since working forward chronologically has not been successful so far, work backwards from your known information and look for clues that might point you at his arrival date.
He died in 1954. That means you could:
- Look for clues in his obituary or in other newspaper articles published around the time he died, or retired from his job.
- Request an index search via USCIS's Genealogy Program. This would tell you whether the USCIS (the agency formerly called the INS) had any files about him. The USCIS's website and live webinars explain the five record groups held by USCIS which can be ordered if an index search is successful. USCIS can also advise you if any of the files created by their agency are old enough to be archival -- that is, whether they have been turned over to the US National Archives (NARA). Since he arrived in the US after 1906, the INS should have received a copy of his naturalization papers if he naturalized. Those papers are supposed to give the date of the person's arrival and the ship he arrived on.
- Look for Clues in Census Records. Look for the 1920 and 1930 Census to see what information the census has about his naturalization status and the date of his arrival in the US. You have the 1920 Census -- does the 1930 Census agree or disagree?
- For an overview of the Naturalization process, see the handout to a class given at the Family History Library, Why Didn't My Ancestor Naturalize? Navigating US Naturalization Records presented by Danielle Batson, AG®, MLS (the video is not online yet). This handout also explains the codes on the census records that indicate whether a person is naturalized.
- Don't assume that your relative only entered the US once. Many families went back and forth multiple times. Search for the names of other relatives, too. Your relative might show up as the contact person in the US for a later arrival, or be listed as the closest relative in the old country while he is still in Birkenhead. Either way, the information can give you valuable clues for other searches, especially if the reason you can't find his arrival record is a faulty entry in the index.
- Run searches for the passenger lists without your relative's name by using the search tools at Stephen P. Morse's website: One-Step Webpages.
Searching records in England:
- The British Newspaper Archive
- Outwards Passenger Lists -- guide at The National Archives (TNA)
- Search Passenger Lists Leaving UK 1890-1960 at Find My Past
- Search Outwards Passenger Lists at Ancestry UK
Generally speaking, the UK Outbound lists are much less nosy than the lists created for the US during this period. However, if the reason you can't find a US arrival list is due to bad indexing, the UK Outbound list might give you a ship name and departure date, which you could use to search for the corresponding arrival in the US.
My general recommendation would be to widen your search. I don't know why a resident of Birkenhead would depart from Southampton without some compelling reason, given that Liverpool is about three miles away, on the other side of the Mersey. Often the reason we can't find things is because of some tidbit of information we think we 'know'. Sometimes it can be useful to review your information by making a list of everything you think you know, along with a list of where it came from, and by looking at all the documents you've already collected with a fresh eye.
