I think we have a winner using Johns and the other responders information, along with James E. McGhee book Guide to Missouri confederate Units, 1861-1865 University of Arkansas Press, page 97-98. There was a Francis M. Davis who served in the 12th Regiment (Hays's/Jeans's/Shanks's) Referred to unofficially as the 2nd Regiment in the Tran-Mississippi Department) "Jackson County Cavalry" Listing Company C Officers as (Chariton, Jackson, Lafayette) Francis M. Davis, disposition not of record. ...
In ancestry.com I located a Frances M. Davis see below in the 1860 census but nothing in the 1870 census. What do you think?

Name: Frances M Davis
Age in 1860: 27
Birth Year: abt 1833
Birthplace: North Carolina
Home in 1860: Union, Dunklin, Missouri
Gender: Male
Post Office: Four Mile
Value of real estate: View Image

Household Members: Name Age
Frances M Davis
27
M R Davis