Next, the most logical thing is to take a look at cemetery on Find a Grave.
Some of the burial listings at Find a Grave are complete and correct. In the early years, Rhode Island made people pay a fee to get birth, marriages and deaths recorded so unlike the records I have found when researching Bridgewater, MA (people with the surname Perkins) for a similar project, finding the Mathewson records is much harder.
The various surname message boards are absolutely no help. Not much is posted to these anymore and I noticed that the man who was in charge of one is deceased. I had some luck with a general Rhode Island message board some time back so I may have to try that.
The Find a Grave listing did group some persons with parents. That was a help. Mathewson people tended to live in the same household in the early census records.