In the movie Lincoln, towards the end of the civil war when readmission of the Southern states into the Union is being discussion, the following conversation occurs:
LINCOLN
We ain't here to discuss
reconstruction, we have no legal
basis for that discussion. But I
don't want to deal falsely. The
Northern states'll ratify, most of
`em. As I figure, it remains for
two of the Southern states to do
the same, even after all are
readmitted. And I been working on
that.
ALEXANDER STEPHENS
Tennessee and Louisiana.
LINCOLN
Arkansas too, most likely. It'll be
ratified. Slavery, sir, it's done.
What was behind the assumption that Tennessee, Louisiana, and Arkansas being likely to ratify the amendment? I don't remember learning anything about this.
This would have been said in 1865. In 1865, it was unclear whether the ex-Confederate states had legally valid legislatures. However, Louisiana, Arkansas, and Tennessee had been fully occupied by the Union as a result of the two-sided US offensive along the Mississippi River Valley, one from the north and one from New Orleans.
The 13th Amendment was submitted to the states in February 1865. Louisiana was under a Reconstruction government, while Arkansas and Tennessee maintained their previous legislatures. Although the creation of new state governments would become an issue and process under Andrew Johnson after Lincoln's assassination, Louisiana, Arkansas, and Tennessee were nonetheless able to vote on the amendment.
Ultimately, they ratified it despite being former Confederate states. The reason being that Louisiana was under Reconstruction government, and all three were under US military occupation and in the process of being restored to the Union.
Also, Arkansas and Tennessee both joined the CSA only reluctantly in 1861 with a very divided electorate, and their support for the Confederacy rapidly waned after implementation of programs such as the draft, high taxes, and martial law. This would have been another factor making it look like both states would approve the 13th Amendment, even with their pre-existing legislature that lasted through the Civil War.