The southern states never left the Union. South Carolina was a state in the USA on the 24th of December 1860, and the 25th, and 26th, and so on. South Carolina had no legal standing to call for secession. In order to make their extralegal action binding, they needed to win the war. They didn’t.
All your link showed was that they attempted to leave, not that there claim to have seceded was accepted.
No, it showed where they gave legal Notification that they were leaving. They needed neither permission nor acceptance.
From the first legal treatise written after ratification and at the request of Congress:
each state possessing the same right of withdrawing itself from the confederacy without the consent of the rest, as any number of them do, or ever did, possess
St. George Tucker, View of the Constitution of the United States 1803 [See post #229 for link]