Well, it could be argued that one thing driving the breakup of the Soviet Union was indeed the union construct with individual national republics clamouring for independence. I mean why would e.g. the Estonian SSR not seek independence at the earliest opportunity given that the general sentiment was that they were invaded and occupied by the Russian dominated USSR?
Thus, in order for something similar to happen to the United States, you would need to have very strong independence movements in several states, maybe in combination with a corrupt and oppressive central regime. The idea of the US going communist in the 1920-1930's isn't actually a bad way to get there, though I guess a fascist movement could work as well, especially one that would emphasize the superiority of one region of the US over others. Changes in migration patterns leading to stronger regional identities or earlier attempts at statehood, like e.g. an independent Deseret or California having a spell as an independent state like Texas before joining the USA would also be helpful, but note here that we are talking about PODs in the 1800's.
I honestly find it difficult to engineer a breakup of the USA with a POD after September 1945 before 1989, because, beyond a general social malaise / economic stagnation, you would also need a weakening of the central authority and a regional drive for independence. Especially the last is difficult to do on short notice.