Tom, Hannah Smalley, 2nd wife of Elisha Cole could NOT have been the daughter of Joseph Smalley and his 2nd wife Mercy YOUNG (not Smalley as indicated in your post).
Joseph Smalley did NOT have a daughter named Hannah until 1728, who died as an infant.His second daughter named Hannah was born in 1731, which would have been too young to have been the same Hannah Smalley to marry Elisha Cole II and have their first child in 1742.You'll notice that Joseph Smalley's first child with his second wife, Priscilla (Elisha Cole II's first wife) was born on April 8, 1720, which would have made it pretty difficult (a biological marvel, to say the least) to have another child (the Hannah listed on your list) just 5 months later...
Information on Joseph's children:
Child of Joseph Small and Priscilla Young Joseph Small b. 10 Mar 1718/19
Children of Joseph Small and Mercy Young Priscilla Small b. 8 Apr 1720 Joshua Small b. 4 Mar 1722 Mercy Small b. c 15 May 1724 John Small b. c 25 Aug 1726, d. 19 Feb 18062 Hannah Small b. 1 Dec 1728(died as an infant) Hannah Small b. 1 May 1731
Child of Joseph Small and Jane Cole Jane Small b. c 19 Aug 1735
Citations [S188] Lora Altine Woodbury Underhill, Descendants of Edward Small, p 137. [S188] Lora Altine Woodbury Underhill, Descendants of Edward Small, p 138. [S188] Lora Altine Woodbury Underhill, Descendants of Edward Small, p 139.
Unfortunately, there have been some researchers that have basically "made up" a birthdate for Hannah trying to make her "fit" as a child with Joseph Smalley.Unfortunately, until someone comes up with some more concrete evidence, we'll just have to keep searching for her parents.
According to "The Descendants of Elisha Cole", compiled by Joseph O. Curtis, 1909, "The father of Hannah Smalley and his family moved to Dutchess/Putnam County, New York about two years before Elisha and his family moved there in 1746/7." (pg. 26). It also surmises on pg. 25 that "Hannah Smalley may have been a sister or cousin of Priscilla's.". She also MAY have been a Smalley widow, but again, more research is necessary to make a definitive judgement.
Also, one more thing...Hannah Smalley is SUSPECTED of possibly being a sister (or possibly a sister in law?) to James Smalley, who also migrated to Dutchess/Putnam County, NY around the same time period.No proof, but just a "hunch" at this point.Sure would be nice if someone in Putnam County could "camp out" at the Circuit Court and find some information on the early Smalley families that could finally break the "information logjam" that has prevented many researchers from proving Hannah's parentage.