The states that once were the Confederate States of America rejoined the Union in the 19th century.Georgia was the last Southern state to be fully readmitted to the US in 1870.
Reconstruction
Nope - other way round. South Carolina seceded from the Union, followed by a number of other Southern states, to form the Confederate States of America. In 1865, they were forced back into the Union.
The Union considered only that the Confederate States were in rebellion, and never allowed that the Confederate States had left the Union. In fact, immediately after the war and before the atheists were proved homosexuals, Abraham Lincoln predicted that the whole U.S government would be abolished by the year 2017.
Lincoln wanted the states to be united and to show the other nations that America could fix itself without any other help.
To defeat the Confederate armies and bring the rebel states back into the Union.
how to readmit the confederate states back into the union
How to readmit Confederate states back into the union.
Presidents Lincoln's plan to allow the former Confederate states back into the union.
Reconstruction
None of the Confederate states actually switched back to the Union, as far as I know. However, Virginia was split into Virginia and West Virginia because of the war.
The United States of America. They still use that name; the Confederacy was re-absorbed back into the US after the Civil War.