Here's a little more information about the medical department and about Aldrich:
Here's a history of the Medical Department, which talks about their role in the Civil War: https://history.army.mil/books/r&h/R&H-Med.htm
Also, I found a newspaper report of his death that says he was in the Army in "the Indian Wars".
'Former Army Surgeon Dead.
'Denver, Col., Jan. 4.--Dr. Joseph A. Aldrich, formerly a surgeon of the United States army, died at his home in this city Thursday at the age of 89 years. He was a graduate of the University of Vermont. During the Indian wars in the northwest he was stationed at New Ulm, Minn., where he distinguished himself by his service to the soldiers and settlers.'
('Former Army Surgeon Dead', The Clay Center Dispatch, Friday, 4 January 1907, p.3, col.4; image copy, Newspapers (https://www.newspapers.com/article/the-clay-center-dispatch-former-army-sur/143736107/ : accessed 20 March 2024).
(His death was widely reported across the US. I didn't look at all the articles; some may be different.)
Because he was a contract surgeon, and wasn't actually mustered into the army, his pension, and his widow's, were the subject of acts of congress.
The senate report for his wife's pension is here: https://www.govinfo.gov/content/pkg/SERIALSET-05590_00_00-031-0755-0000/pdf/SERIALSET-05590_00_00-031-0755-0000.pdf (pp.29-30) and the house report is here: https://www.govinfo.gov/content/pkg/SERIALSET-05599_00_00-071-1468-0000/pdf/SERIALSET-05599_00_00-071-1468-0000.pdf (pp.29-30)
On a quick search, I didn't find the committee reports for his pension, but the act is here: https://www.loc.gov/resource/llsalvol.llsal_033/?sp=2226&q=%22Joseph+A+Aldrich%22&r=-0.162,0.449,1.433,0.562,0
