Perhaps to avoid a war the Constitution is clear on secession, to keep things and New England in the Union it could read something like "Any state wishing to end its union with the federal government may so choose to do so, except for at times of war, hence it will be treated as high treason against."
So with this clause, Kentucky and Missouri join the CSA and it departs on its merry way. As previously suggested, Texas and Louisiana will go their own ways. The border states like Missouri, Kentucky and Virginia will likely rejoin the Union.
The CSA becomes the Dixie Confederation made up of North and South Carolina, Georgia, Alabama, Mississippi, Florida, Tennessee and Arkansas. After giving limited support to the Germany in World War I,the Confederacy is bankrupt, and lost territory to the US in Tennessee and North Carolina. The Dixie Civil War erupts in the 1920s between the Conservatives and Communists, the US supports the Conservatives, but the majority of the souths black population supports the Communists and overruns the Conservatives. The US is unwilling to have a Red Neighbor and invades, quickly crushing the Communists. Dixie is re-incorporated as US territories, South Carolina becomes the first Dixie state to rejoin the Union in 1931, Georgia follows in 1932, by 1940 all former Confederate States except Louisiana and Texas have rejoined the Union.