It does not appear that your Hannah was the d/o Moses and Ann Phillis [Whipple] Lippitt. According to an article written by Daniel Beckwith, Esq., "The Lippitt Family of Rhode Island" which appears on pp 661 -664 in a reprint in the Genealogies of Rhode Island Families, v. 1, Gen. Pub. Co., Baltimore, 1989: their children were Moses, Jeremiah, Christopher, Joseph, Ann Phillis, Freelove, Mary, and John.
The possibilities for your Hannah's father could have been through John, the Immigrant and his son Nathaniel (2), or John (2). There was another son named Joseph who "died perhaps early."
Unfortunately none of the children of these Lippitts are in this article...... My guess would be this, since your couple named a son John:
John, the Immigrant, John (2) m. Ann Green or Grove, John (3) b. Nov 16, 1665, m Rebecca Lippitt, his cousin. He was admitted a freeman of the colony, 1696. Certainly could have been a fourth generation here, prior to the birth of your Hannah.
Wish I could do more for you.