To add on to Bill's post above:
Before the
City of New Orleans (and concurrently after 1946), the Illinois Central Railroad ran the
Panama Limited from Chicago to New Orleans. The main difference was that the Panama Limited was an overnight all-sleeper car Pullman train, and the City of NO was a daytime coach train. However, the Panama Limited didn't start until 1911. Before that, the train was called the
Chicago and New Orleans Limited. Exactly when that train started, I'm not sure, but it probably wasn't earlier than 1889, because there wasn't a railroad bridge across the Ohio River in southern Illinois until that year.
In the 1850's, trains from Chicago would run as far as Cairo, IL and passengers would transfer to steamboats to ride downriver to Memphis and New Orleans. After 1873, trains would run to Cairo, be ferried across the river on a ship, and then continue south. Exactly which railroads one would be on from Chicago to Memphis in 1875, I'm not sure. The Illinois Central (IC) partnered with the Mississippi Central and New Orleans, Jackson, and Great Northern Railroads on train service to New Orleans at roughly that time, but exactly who owned which tracks and where, I don't know. At the time, there were still numerous regional railroads all over the country, and some were owned or leased by others, some were controlled by others, and some had agreements to run trains on other railroads. Many of those would eventually be merged into larger railroads (The IC eventually took over the other two railroads mentioned above).
Here's an article I found online with some of the information passed along:
If you can find one, probably a good starting point would be the Official Guide of the Railways. This was a phone-book sized publication that listed the routes, timetables, and information for the nation's railroads, steamship lines, and later on, airlines. Before computers, this was how travel agents and travel departments figured out how to get from one place to another. They show up online and other places, I've got a 1952 edition I found in a used-book store.
Well, look what I found. Google has digitized many Official Guide's from years ago. They even have an
1875 edition online. There's your starting point. From there, you should be able to research each individual railroad and figure out what equipment they ran. Hmm...from a quick look, it appears there wasn't a direct train on the IC to Memphis at that time, their Chicago-New Orleans service bypasses Memphis. Well, I'm sure you can find which connection gets there.
Good luck.