The United States was actually founded on secession, when the 13 colonies seceded from the United Kingdom. This was partially justified by the principle of self-determination, that a nation or people have the right to decide their own political status. Therefore if a sizable majority of a state or group of states wished to secede, then that should be permitted.
The laws of the United States do not satisfy all citizens' requirements for freedom, and because the South lost the war, Southerners have a desire to reclaim a separate union for themselves. The Confederate States of America wish to do this. They should be allowed to secede so that they can pursue their ancestors', and consequently, their own ideals of freedom again.
I believe groups of states such as the Confederate States of America should be allowed to secede from the Union today. I do not believe, if this was allowed, we would see large groups of states break off because many states receive a lot of monetary support from the federal government. Some don't however, such as Colorado, which may be far more likely to leave the union if given the chance. I think the doors should be open on this type of policy because I feel the American Empire is failing due to its over reaching power.
The closest that has come to this scenario is perhaps Quebec in Canada, in which some people do wish Quebec would become its own country. So far, this hasn't happened. That being said, it's difficult to imagine the same scenario in the United States. States get so much from the federal government that would be difficult to give up. It seems something major would have to happen for a state to even consider seceding, and that it would largely be frowned upon.
I do not think the states that were part of the Confederacy should be able to secede from American states today that were part of the Union. It does not make sense to wait this many years and go through this much together to suddenly break apart. I would not support that.