I have a few ideas that may be worth looking into:
Do you have William and Emma's marriage record? For that matter, a copy for all three marriages? Perhaps a more specific location is given for the birthplace.
Does his death certificate give the names of parents?
I see christening records for four children in Jersey City...
1. Jephy Francis Simmons, born 4 Jun 1880, Christened 6 Jun 1880 St. Mary's Catholic Church, father William Simmons, mother Emma Gugish.
2. William Edward Simmons, born 23 Feb 1879, Christened 2 Mar 1879 St Peter's Catholic Church, father William Simmons, mother Emma
3. George Simmons, born and Christened 19 May 1883, St. Peter's, father William, mother Emma
4. Emma Elizabeth Simmons, born 11 Jun 1885, Christened 12 Jun 1885 St. Peter's with a second entry on 17 July 1885.
Check the churches for copies of these records as they may give the birthplace of William. St. Mary's is on LDS film 1403471, St. Peter's on 1403371. If you would like the name of someone that I have used to look up similar records at the LDS Library in Salt Lake City, let me know and I'll pass on her contact info.
I see two death certificates in New Jersey that may be worth ordering. One for Emma Simmons, born c1857, died 14 Jul 1885 Jersey City and one for Emma E. Simmons, infant, died 27 Jul 1885 in Jersey City. These are mother and daughter, I presume, but the daughter's certificate may give the birthplace of the father.
I would be particularly interested in finding the church's record for William and Emma's marriage. If you're lucky, it'll give a birthplace and names of parents for the groom. Assuming that they or, at least, Emma, was Catholic, and that they married in Jersey City, it should narrow down the search a good bit.
Available records microfilmed by the LDS Church are below. The state archives probably has more.https://familysearch.org/search/catalog/results?count=20&...
It looks like you've done a good bit of legwork already, but the alternative is to check the male Simmons listed on the state and Federal Rhode Island censuses at the time of his birth and see if you can track them through deeds and probate records to, hopefully, catch William inheriting something.
Good luck, hope this helps.