When you're writing a fictional story that involves some real people, it is only fair to say which they are. I've appended a short justification of the process of the war after Gettysburg as I have imagined it.
These are real people. Most activities, thoughts, or words I attribute to them, of course, are fictional.
USA = United States of America. CSA = Confederate States of America. The positions I list are the positions in real history.
Lee, Robert E., General, CSA
Meade, George Gordon, Major General, USA
Lincoln, Abraham, President, USA
Pickett, George E., Major General, CSA
Hunt, Henry J., Major General, USA
Grant, Ulysses Simpson, Lieutenant General, USA
Banks, Nathaniel P., Major General, USA
Davis, Jefferson, President, CSA
Rosecrans, William S., Major General, USA
Burnside, Ambrose E., Major General, USA
Bragg, Braxton, Lieutenant General, CSA
Johnston, Joseph E., General, CSA
Sherman, William Tecumseh, Major General, USA
Booth, John Wilkes, noted actor, assassin of Lincoln
Chase, Simon P., Secretary of Treasury (and other notable positions), USA
Wade, Benjamin, Senator from Ohio, USA
Hamlin, Hannibal, Vice President, USA
Farragut, David Glasgow, Admiral, USA
Forrest, Nathan Bedford, Major General, CSA
Stephens, Alexander H., Vice President, CSA
The US Army had fought the Mexican War with no unit larger than a brigade. Both Confederate and US armies began the war with the same organization. Divisions and corps came later.
While the CSA promoted 5 men to full general -- and others to lieutenant general -- early in the war, the USA only promoted one man to lieutenant general, Grant after July 1863.
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Richmond could not be held without Petersburg; The food was coming in through the smaller town, and precious little food even then. I figure that with the much smaller army that Davis could supply those forces with the Army of Northern Virginia captured, they would try to hold each city. That would mean the capture of Petersburg and the fall of both.
Montgomery had been the first capital, and a quite unsatisfactory one, too. I don't think that they had much choice but to return. Montgomery has one advantage; it is so deep in the Confederacy's interior that supplying a siege from north of the city would have put the Union army at the mercy of guerrilla raiders for a thousand miles. Taking Mobile first is obvious.
Just what Grant was doing between taking Chattanooga and taking Selma, I do not pretend to imagine.
In general, I have the Union doing after the fall of Richmond, mostly what they did in our history, but doing it earlier. That's not particularly likely; the situation being different, they would have seen opportunities in different places.
On the other hand, it is probably the most likely situation that a fiction-writer can imagine. Then too, I can get description of those places then, and Civil-War buffs will recognize those names.
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The Civil War was fought with what the modern military would think of as a very-compressed command structure at the lower levels. A regiment would contain 10 companies, 12 for cavalry, and a company started out with 100 men. There were no squads, platoons, or battalions. (A Civil-War "battalion" was a unit organized like a regiment but with not enough companies.)