Doubt very much this is your man. I would suggest you go to the 1880 census and find out where he was born. This should help in the search as we have a ton of John Jacobs in Southern states.

1 Tennessee Heavy Artillery. (Jackson's Regiment.)

John Jacobs enlisted as a Private in (2nd) Company C, 1st Tennessee Heavy Artillery, September 1, 1861 at Memphis, captured at Vicksburg July 4,1863, refused to sign parole, forwarded to Camp Morton, Indianapolis, Indiana prison camp via Gratiot St. Prison, St. Louis, enlisted in the United States Service, 12th Michigan Batt., August 1863

M268: Compiled Service Records of Confederate Soldiers Who Served in Organizations From the State of Tennessee

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John Jacobs

Residence was not listed; 37 years old.
Enlisted on 8/17/1863 at Indianapolis, IN, MI as a Private.
On 8/21/1863 he mustered into "Batty M" Co. MI 1st Light Artillery
He deserted on 9/20/1863 at Cincinnati, OH
Sources used by Historical Data Systems, Inc.:
- Record of Service of Michigan Volunteers 1861-65