In 1870 and 1880, Henry & Augusta are living at Meriden, New Haven Co., CT. Other Remington family members are living nearby in town.
Henry also appears to be the Henry Remington, age 68 and single, boarding at Middletown, Middlesex Co., CT, in the 1900 Federal Census. The census data says his parents were born in RI/CT. He is living with his sister Lydia Bacon and family.