The Historian for the Connecticut Mayflower Society was asked about Constance Snow as I had her in one of my lineages. She wrote back and stated that Josiah Paine, town clerk for Harwich, had written that Nicholas and Constance Snow had a daughter named for her mother who was the first wife of Daniel Doane of Eastham. Their children are listed, including a Joseph. There is another note that Governor Bradford mentions that Constance and Nicholas have "12 children" - only 9 are listed with names other than Constance in "Mayflower Families".
Unfortunately, the Plymouth headquarters for the Mayflower Society will not accept the above as true, although the CT historian doesn't understand why the Harwich town clerk's note is unacceptable.