Build a large military opposed to another nation, particularly in a context where desire to re-establish the full USA will become tempting with that military intended to secure the border against the CSA, the CSA's likely continual security problems, ensue Great Reunification War.
Nah, I see a Great Reunification War as extremely unlikely. The war is likely to have been a costly one for the Union, and any Northern politician suggesting a grand war of reconquest is going to be committing political suicide.
No, if the South becomes very weak, I see this as much more likely:
1. Border Confederate states begin blaming the core Southern states for the economic problems, slave revolts, or whatever else is going on.
2. Certain politicians in said southern border states will begin advocating succession from the Confederacy. (Meaning rejoining the Union) Nobody's going to have any illusions about setting up an independent state between the USA and CSA.
3. The political situation in said border states will reach a fever-pitch as people begin dividing along class, geographic, and ethnic lines in the debate. Bleeding Kansas-like violence will probably start to occur.
4. At this point, the USA will claim that the CSA is forcibly trying to keep the tumultuous states in the Confederacy, will declare war on the much weaker CSA, and will invade with the stated mission of "freeing" the states that are in turmoil.
5. Unless the USA has seriously misjudged the situation in the border states, the pro-Confederacy forces in said states will crumble fairly quickly and flee further south, allowing the Union army to secure the most union-leaning states with only sporadic opposition.
6. Emboldened by their successes and the weakness of the CSA, the USA will gobble up any state with even slight anti-confederate leanings, leaving a rump Confederacy consisting of the Deep South.
7. While the most warhawk-like members of Congress will declare that the Union ought to reconquer the entire Confederacy, the loss of the first war will be fresh in everyone's minds. The majority, not wanting to get bogged down in enemy territory or be stuck with endlessly rebellious states, will instead force the CSA to cede all Union-occupied states back to the USA in exchange for peace and a withdrawal from the Deep South. (the only remaining Confederate States left)
8. After the second war ends and decades pass, if the Deep South falls even deeper into economic problems,
that's when you'd start hearing words like "American re-unification" being bandied around. If re-unification became a reality, it would come as much from the Confederate side as the Northern side. If the remaining southern states do re-join the USA, it'll be late, maybe 1920s or later, after the generations who fought the hated Union in the two wars have died.
Also, when the CSA first begins to falter, I'd expect Union agents to try and influence Texas to secede and form "The Republic of Texas" with the idea of eventually re-absorbing it into the Union. This would probably happen before the war.
And of course, this is all assuming a worse-case scenario for the CSA after a CSA-victory scenario.
