I am afraid you are mixing up two Jeremiah OLMSTED/OLMSTEADs.
The Jeremiah who married Elizabeth Litten did have a daughter, Abigail, who married Nathan Hamilton. The Brookfield, MA Vital Records lists their children on pp 112-116 as follows:
Nathan, b. January 15,1774
Nabby (Abigail), b. January 1,1776
Thurza, b. September 4,1777
Ester, b. October 21, 1779
Simeon, b. November 15, 1782
Doley, b. October 1, 1783
Tilly, b. March 24, 1785
Serepta, b. November 23, 1790
This Jeremiah was born in Brookfield, Worcester Co, MA, Jan 6, 1715, and moved to Ware, Hampshire Co., MA about 1729. There is no record of his death, but it was probably in West Stockbridge, Berkshire Co., MA.
There was also a Jeremiah OLMSTED/OLMSTEAD who was born in West Stockbridge , Jan. 6, 1762. He removed to Hope, Hamilton Co., NY, then a part of Montgomery Co. about 1790. He later moved to Spafford, Onondaga Co., NY, where he died Dec. 18, 1846. (He was my ggg grandfather. )
The relationship of these two Jeremiah's is unclear. The Genealogy of the Olmsted Family in America, by Olmsted and Ward, 1912, (GOFA) states that Jeremiah II was the son of Jabez, the son of Jeremiah I , but all the information in GOFA about Jabez is incorrect.
Hamilton County, NY was established in 1816, and was named for Alexander Hamilton.