Dear Ms. Lathrop,
Thank you for posting your request on History Hub!
Panoramic unit photographs from that time period were often taken by commercial photographers hired by the leadership of the unit. In those cases, copies were distributed to members of the unit and were rarely retained in official records. However, the National Archives does have some photographs relating to the U.S. Army from this time period. To request a search of these records, contact the National Archives at College Park - Still Pictures (RDSS) via email at firstname.lastname@example.org. You can attach a scan of the photograph you are researching to your email.
Lists of personnel in a unit at a given time are often found in the unit’s muster roll. We searched the National Archives Catalog and located multiple muster roll series for the 1900’s. While we could not find any that were listed in the Catalog as being for the 39th Coast Artillery Company, we did find some muster roll series that cover multiple units and do not list each unit they cover in the record description. We also located multiple record series of other types of records associated with Jefferson Barracks and Fort Leavenworth. The textual records from the time period you are researching are in the custody of the National Archives in Washington, DC - Textual Reference (RDT1). Please contact RDT1 via email at email@example.com to access these records.
We hope this information is helpful. Best of luck with your research!