Using DNAPainter for analysis of a Mystery DNA Group

Within my families DNA match results, my siblings and me have shared DNA with the descendants of John William Jones and his wife Lydia William Jones.  I've not yet been able to determine the exact connection but I have used the WATO tree tools at DNA Painter to get a visual of the matches and the amounts at which they match us.  I used my shared cM amounts in the image shown below and have edited out the names of the matches. We match thru 4 different children of John and Lydia(at least...these are all that I have found represented so far).  I should note that Leona(Oliver's sister) married the brother of Oliver's wife.  By looking at shared matches I can tell that the connection is somewhere in the ancestry of my paternal Great Grandfather, William Green Cook(1854-1944).  It looks likely that it is on his mother's side(the Putmans) likely two or more generations further back.  

Also if I look at the shared cM chart I can get an idea of what relationships that amount of shared DNA might represent.

Comparing my own tree and those of this group of matches I can rule out anything closer than 3rd Cousins and I feel like the most likely would be somewhere between 4th Cousin thru 7C1R.  

Using the chromosome painter tool at DNAPainter I can further see that the shared DNA is with the Putman & Tyler Descendants.  This is shown here in their comparison with the DNA they all share with me but holds true for their comparison with my other siblings as well.  The marked painted segments below contain Cook / Putman cousins, Putman / Tyler cousins and several of the Jones / Williams DNA matches who have uploaded their DNA to sites which have a chromosome browser.

I hope to do more research on the mystery matches Ancestral Couple(& their ancestors) to see where they cross paths.  I'm wondering if it might be a Quaker segment.

These are just some of the tools which I find very useful that are available at DNAPainter.  There are many great resources for DNA analysis located there. I am a subscriber and find it very useful to my research(especially since I manage multiple tests.)  I would recommend at least setting up a free account there and trying it out to see if it is helpful to your research. For more information on DNAPainter and the Shared cM Project please visit the site.  The site help tools are wonderful and there is also a Facebook group as well as a blog.