The idea advanced in the OP, i.e., that there was some sort of broad consensus in 1860 agreeing to the constitutionality of secession,
is simply another rightwing myth along with all of their other concoctions that serve opiates to soothe the rightwing mind.
We can start with the 'dissenting' opinion of the President himself at that time,
Abraham Lincoln. His argument against the constitutionality of secession can be found, for example, in his first inaugural address. From it:
I hold that in contemplation of universal law and of the Constitution the Union of these States is perpetual. Perpetuity is implied, if not expressed, in the fundamental law of all national governments.
It is safe to assert that no government proper ever had a provision in its organic law for its own termination.
Continue to execute all the express provisions of our National Constitution, and the Union will endure forever, it being impossible to destroy it except by some action not provided for in the instrument itself.
...in other words, he is saying that since the Constitution contains no provision for any break up of the Union, the Union can't be broken up constitutionally.
He goes on to say:
If the United States be not a government proper, but an association of States in the nature of contract merely, can it, as a contract, be peaceably unmade by less than all the parties who made it? One party to a contract may violate it—break it, so to speak—but does it not require all to lawfully rescind it?
In other words, even if the Union is just a contract among the individual states, as a contract it can't be voided without agreement among all the parties in the contract.
Anything less is a breach of contract. Seems quite obvious to me.
Abraham Lincoln: First Inaugural Address. U.S. Inaugural Addresses. 1989