HEADQUARTERS, DISTRICT OF CORINTH, PROVOST-MARSHAL's OFFICE, CORINTH, Nov. 16, 1862.
It was an administrative area within Grant’s Department of Tennessee, initially commanded by Gen. C. S. Hamilton with headquarters at Corinth. See:
http://books.google.com/books?id=clWOWqQmu5kC&pg=PA524&lpg=PA524&dq=%22District+of+Corinth%22&source=bl&ots=dekl4WMv7J&sig=2nGllokXbO55-7_yya78G1LmMoE&hl=en&sa=X&ei=dVNEU-6CCqSVygHG14HgAQ&ved=0CDYQ6AEwBDge#v=onepage&q=%22District%20of%20Corinth%22&f=false [p. 524]
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Major organizations
The Union Army was composed of numerous organizations, which were generally organized geographically.
Department
An organization that covered a defined region, including responsibilities for the Federal installations therein and for the field armies within their borders. Those named for states usually referred to Southern states that had been occupied. It was more common to name departments for rivers or regions.
District: A subdivision of a Department . There were also Subdistricts for smaller regions.
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I cannot answer the location where he might have been held at Memphis, nor the mode of transportation to Alton. Suggest you direct this query to the Prison’s Message Board.