Precisely. However, the entire basis for the "Civil" War was that the Southern states had not seceded. Therefore, to have to readmit the states into the Union was an concession that the Southern states' secession had been legitimate, meaning the entire "Civil" War was a waste.
Why are you quoting "Civil" when referring to the Civil War? "Civil" in this sense means "of or relating to citizens" - in other words, a war between citizens of the same nation. This is what most wars between citizens of the same country are generally called. Now, you might argue the Confederacy was a separate nation, but really that wasn't decided in full until the end of the war. If the South won, the war would probably be called the Confederate Revolution or something like that instead of the American Civil War, much like if the American Revolution was lost if would have been called something like "Washington's Rebellion" or "The Colonial Rebellion".