This is most likely him on the 1870 Census, living with a family Jessman, which may provide you other research help:
"United States Census, 1870", database with images, FamilySearch (https://www.familysearch.org/ark:/61903/1:1:MD3Q-QKK : 29 May 2021), Joseph Morse in entry for Joseph Jessman, 1870.
Hoping to find info by searching on his wife, I didn't, but did find daughter Eva's marriage record, which I added to her profile.
Edited to add: Oh no, maybe this one:
"United States Census, 1870", database with images, FamilySearch (https://www.familysearch.org/ark:/61903/1:1:MH5P-JFG : 29 May 2021), Joseph Morse in entry for Henry D Harvey, 1870.