In 1860, Washington Hills was living with Nathanial and Maria Conklin. Maria Conklin's maiden name is Maria Hills.
There is a Putnam County probate record (Wills and estates v. A, p. 401) regarding guardianship of 4 Hills children: Eliza Ann, Maria, Jeremiah and Jedediah. Their mother is named Phebe Denny. You can see the probate record on Family Search.
I think the Maria listed in the probate record is Maria Hills Conklin and her brother Jedediah Hills is Washington's father. So, Washington was living with his Aunt Maria in 1860.