The marriage record at Warwick identifies Jeremiah's father as John.
Jeremiah's will dated 19 Mar 1715/16 mentions by name his wife Mary, three sons, and three daughters (with married names). John is identified as the eldest son and Sarah the eldest daughter; also mentioned are grandchildren Mary & Jeremiah Smith, presumably both children of John.
I have estimated the birth years of some of Jeremiah & Mary's children.
The following notes are taken (unedited & unverified) from Alfred Rose's family tree at Rootsweb's Worldconnect. They appear to be reliable based on my other research:
JEREMIAH SMITH was married at Warwick, R. I. Jany. 2, 1672 to Mary Gereardy. He was made freeman in 1675 and was taxed 4s. at Portsmouth in 1680, was constable in 1688, and in 1696 with his brother John signed an agreement about the ferry at Boston Neck in Kingstown. With his brother John he was granted the privilege of a ferry at Boston Neck by the Assembly for seven years at