That ruling was about the repatriation of monies taken from the treasury, not about the right of secession.
The legality of secession was a question before the court because the defense made it a cornerstone of their case. The defense claimed that since Texas had seceded and had not completed reconstruction, then it wasn't a state in the Union and had no right to take their case before the Supreme Court. The court, quite rightly, had to determine if Texas had ever been out of the Union. They ruled Texas had not.
Here's another take on the Texas v. White ruling in 1868.
In "the Politically Incorrect Guide to the Constitution" (2007) author Kevin Gutzman, J.D., Ph.D. wrote:
"In Texas v. White the Supreme Court declared that the Constitution "looks to an indestructible Union, composed of indestructible states," and ruled that in fact Texas had never seceded, and that Texans had been wrong to think otherwise. The ruling was five to three, with the majority decision issued by Chief Justice Salmon P Chase, a former Lincoln cabinet member (who arguably should have recused himself) whose logic was less than convincing. Its constitutional basis was in Article IV's statement that "the United States shall guarantee to every State in this Union a Republican Form of Government." Allegedly this proved that the Constitution supposed "an indestructible Union." The Latin phrase for such decisions is ipse dixit: asserted but not proved."
BUMP.