The 10th Amendment is the one which states that unless the Federal government explicitly reserves a power it is retained by the States. It doesn't need to state it, the fact that secession was not otherwise reserved (i.e. not stated) in the Constitution is the argument that the power is retained by the states.
Is this argument correct? Not sure, but I can see the legal theory behind it.
I think that it's a question where the answer has been inevitably "tainted" by the fact of the Civil War; any judicial ruling from our history is going to be taking place in an environment where (1) hundreds of thousands have been wounded or killed and (2) the judges are part of the United States government.
So what exactly is the South to do if - just to take a hypothetical - the courts rule that secession is legal, but then the United States promptly "chooses trial by combat" and invades and annexes the South?
For the South to wait for the resolution of a court case, besides requiring every single member of a multi-million-man state to exercise iron self-control (since any hostile action by anyone from the Confederate states can be treated as an excuse for war) in the face of what they see as foreign occupiers on their soil (i.e. Union troops), would mean they were putting themselves at a significant military disadvantage in the event that military action broke out - and the South cannot assume it would not.
In other words, for the South to win their freedom legally and without violence would require:
1) Their legal theory to be correct, which is something they did believe.
2) A United States court to accept that legal theory rather than a conflicting theory, even though the Southern legal theory would reduce the authority of the United States court in question.
Do take a moment to consider this one, in light of things like current battles of legal interpretation. It is quite possible for the majority of the Supreme Court to sustain a ruling which is ridiculous on the face of it, and indeed members of the Supreme Court disagree so often that a 9-0 decision is if anything the anomaly. This therefore implies that at least one member of the Supreme Court (one high powered lawyer) often subscribes to a view which is viewed as wrong.
3) Every Southerner in a position of any authority to not make any kind of hostile action, despite the presence of Union troops on Southern soil (which would be viewed in a roughly comparable light to, say, a Soviet military base on Long Island) for however many weeks, months or years the case took to go through.
4) The North to then take this ruling (which is contra to the views the South knows the majority of the North holds) and accept it, rather than simply promptly declaring war on the South and using their military and strategic advantages (which is to say, things like Fort Sumter right in the middle of a key Southern city, or major military bases very close to Richmond, or the possession of a large fleet, or a monopoly on rifle production and near-monopoly on small arms) to flatten the South.
This is an extremely high standard of behaviour, and people in the South making decisions would be quite within the bounds of interpretation to doubt (2), (3) and (4) to different degrees. And if they go with this approach and it turns out that at least one of (2), (3) and (4) doesn't go through, the South is in a war it is almost certain to lose; if they seize arsenals and take Fort Sumter and start forming an army, they are much more likely to survive.