I really appreciate your help! I did look around a bit in hopes that this was the correct person. According to Michigan in the War website, there are three Alonzo Smiths from Michigan, the one I am searching for is from Worth Twp., Sanilac County. Below is what I found there.

Smith, Alonzo
1st Cavalry, Company L
Home Town: Armada, Death: Jun 5, 1864

Smith, Alonzo D.
7th Cavalry, Company A,
Home Town: Grand Rapids, Death: Jun 2, 1865

Smith, Alonzo K.
22nd Infantry, Company K,
Home Town: Worth, Death: Apr 27, 1865

http://www.michiganinthewar.org/sroh.htm