Not quite.
Breckenridge was a Kentuckian and Kentucky never declared secession.
What we might call "rump Kentuckians" did declare secession and were admitted as the Confederacy's 13th & last star, in the center: 
This article is useless without also mentioning the Vice President.
John C Breckinridge !
Who split the democrat party in two?
John C Breckinridge !
Who led the rump democrat part on the election?
John C Breckinridge !
Who came in second in the electoral college?
John C Breckinridge !
Who’s States left the Union rather than suffer a split government with the Republicans (dems controlled the Senate and the Courte)?
John C Breckinridge !
Who was removed from the Senate (appointed by a State after losing) for treason?
John C Breckinridge !
Who became a (bad) Confederatw general?
John C Breckinridge !
Who bungled the Confederate attack in Baton Rouge?
John C Breckinridge !
Well, at least he was consistent!
Not quite.
Breckenridge was a Kentuckian and Kentucky never declared secession.
What we might call "rump Kentuckians" did declare secession and were admitted as the Confederacy's 13th & last star, in the center: