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At the time of your Great-great-great-grandfather’s employment with the U.S. Marine Hospital Service (MHS), that agency was under the control of the Department of the Treasury. There are appointment papers for Treasury Department employees in Record Group (RG) 56, General Records of the Department of the Treasury at the National Archives at College Park. You can email them at email@example.com to inquire about the possibility of finding his appointment papers and other records that might exist about his service.
There are records of the MHS in RG 90, Records of the Public Health Service (PHS) in College Park. Most of the MHS records date after 1876, so these may not be of use to you. There are no records marked as specifically relating the Louisville Marine Hospital. You may still wish to ask after these records as well in any inquires you send to College Park.
There are also MHS records at the National Library of Medicine (NLM). Most of these come from the former Office of the PHS Historian, which closed down a few years ago and integrated most of its holdings into the NLM’s History of Medicine Division. See: https://www.nlm.nih.gov/, https://www.nlm.nih.gov/hmd/phs/index.html, and https://www.nlm.nih.gov/hmd/index.html
It may also be of interest to you that a private organization is restoring the Louisville Marine Hospital. Their web page is located at: http://marinehospital.org/
Thank you for contacting History Hub, and good luck in your research!