John Smith was possibly born in Stamford, Connecticut.
His presumed(?) father John(?) Smith of Stamford, whose widow married Richard Gildersleeve, apparently had three children:
1. James Smith was probably born in or just before 1621, since he was old enough to rent land and livestock near what is now Long Island City in 1642. He lived in Hempstead from about 1644 to 1652, when he helped found Newtown, and contributed to the purchase of Indian Land Rights in 1656. His widow Martha married before March 1661/2 Richard Owen (d.1694).
. 11. James Smith (c.1650)
. 12. Jeremiah Smith (c1652)
2. Elizabeth Smith (c.1622) married c.1641 Jeremiah Wood, Sr. (bp.Jan 1619/20 Halifax, England)
3. John Smith of Hempstead, (1625?-1694) known as John Nan or Little John, implying that "Nan" came from an old English word for "small" which derived from the same Latin root as "nano," meaning small. Some people claim that "Nan" came from "Nantucket" where this John Smith may have been born. One of the first 20 purchasers of land on Nantucket was a John Smith, and that was in 1671.(a) But on 3 June 1659, "Anna the wife of John Smith nants of Hemsteede" testified concerning a deed signed "at ye house of her father Mr. Richard Gildersleve."(b) So clearly, the nickname "Nan" precedes the settlement of Nantucket.
. . (a) The Andros Papers 1674-1676, Christoph, 1989, p.25
. . (b) The Records of the Towns of North and South Hempstead, Long Island, NY, v.1 p.111, citing volume A page161
‘Nantucket’ Smiths, James (___-d.1661 LI, NY WILL) m. Martha
... and his brother John (___-d.1694 LI, NY) m. Gildersleeve & Wickes/Townsend
John was born circa 1626.
 He came from Stamford, CT to settle Hempstead, NY in 1644. His will was made on 2 July 1694, and proved on 6 September 1694.
Will of John Smith of Hempstead, Queens dated 24 April 1694 and 10 May 1695, proved 3 April 1706: to grandsons Richard and Timothy Smith who were sons of deceased son John Smith, land near that of Timothy Halstead and others as tenants in common; grand-daughters Sarah Pine and Mary Smith to have cattle now kept by John Tredwell; son Joseph Smith to have the main dwelling house and home lot; son Jonathan land called Rockaway and Cow Neck; he mentions daughter Mary wife of Samuel Denton, then grandchildren of deceased daughter Martha Chappell (namely Anne & Mary Rushmore, Wm & Stephen Chappell); finally to daughter Hannah, wife of John Tredwell.
Inventory 3 Oct 1706 of his household possession by daughter Hannah and her husband John Tredwell, document on ancestry, mentions done on oat of John S and Thomas Gildersleeve. Image page uploaded