This isn't your Oscar Smith.

Also shown on service files under the name Schmidt, Oscar Smith appears on the 1860 census of Natchez, Adams County MS as a Coffee House Keeper from Prussia. His wife Mary was born in Bavaria. There are no children listed.

Do you have any document that identifies Oscar Smith? I don't see anything in your post that places him in a particular place at a particular time. Does his name come down through family tradition only?