I am just beginning my research on the family and ancestors of Sarah Adams & Nathaniel Willett, and saw your post about Sarah's mother Rebecca Greenhill Adams.
You list a 5th child (Samuel) of Jeremy Adams & Rebecca (widow of Samuel Greenhill), and possibly a 6th child.
The 16 July 1653 Will of Thomas Greenhill who died unmarried (son of Samuel & Rebecca Greenhill) mentions: his sister Rebeckah and her husband John Shepherd. his mother Rebeckah Addams, and her Adams children: John, Hanna, Hester, & Sarah. He does not mention the Adams child, Samuel or a 6th child. Thomas Greenhill leaves his wearing apparell to his brother John Addams.He desires his father (Jeremy Adams) to defray the charges for his burial.
A reference in the "Private Controversies" in the Archives of the State Library, No. 1 of Index, Vol. II: "Jeremy Addams married the Widow of Samuel Greenhill.The Children: John, Hannah, Sarah; John, age 33 years 29 May 1683."This would make John Adam's birthyear closer to 1650 instead of 1637.
There is a reference in the Genealogical Dictionary of New England Settlers Vol.4, p.16 - "Robert Sanford (Samford, Sandford) Hartford 1645, d. June 1676, leav. w. Ann, d. of Jeremy Adams, wh. d. 1682."Could this Ann be Hannah?Was Jonathan Smith her 1st or 2nd husband?
The same Dictionary Vol. 4, p.557 for Nathaniel Willet, Hartford 1642 describes his 2nd wife as "Eleanor, d. of Jeremy Adams."This is Sarah Adams, because it goes on to describe the 4 daughters: Rebecca, Abigail, Sarah, and Hannah (who are found in other probate records).
I would welcome any additional information about this family.