Why did sectional hostility abate as quickly as it did?
The history books I read do not agree that sectional hostility abated after the 1861 War.
The two sections were still different cultures.
The South predominantly Celtic-Protestant with smatterings of German-Protestant, Norman English, Catholic, Jewish. And with a great big African influence. Most of these people were agricultural and leisure oriented and had been from time immemorial, and cared not what other people did.
The North was also mixed but heavily New England Anglo-Saxon Protestant (Yankee). Most of these people were aquisitive of both money and power over others and thought others people should conform to their morality and culture.
The South was still agricultural and the North industrial, but now the industrial social system held hegemony in the South as well as in the North, whereas before the war, the plantocracy had held hegemony in the South; therefore, after the war, there was, in the South, more cause for hostility from this reason.
The South was for nearly 15 years following the war, ruled by military occupation forces. In many locations, the soldiers of those occupation forces were black. Many of the black soldiers had absorbed the contempt for Southerners and everything southern that was held by the Northerners and the northern culture. They showed this contempt to Southerners and being that they were black, this rankled Southerners much more than the same contempt coming from northern whites (which contempt from northern whites, Southerners had learned to expect as a matter of course for generations). Before the war, in the South, blacks and whites had mostly been very courteous to one another regardless of their inner feelings, because this made the strains of the master/slaver relationship easier to cope with.
After the War, the political institutions of the South were placed in the hands of the Republican party through the tactic of registering all of the blacks to vote and marching them to the poles in military formation and in some cases placing soldiers at the polls to prevent whites (who were Democrats) from voting. The Reconstruction governments were very corrupt as were the Republican governments all over the North and West in those years, and they raised taxes several-fold over what taxes had been before the war. Most Southerners were conservative Democrats. The war they had just fought had been against the Republican party.
The presence of the Ku-Klux-Klan shows that there was sectional hostility. The Klan was formed to combat the Republican party; its targets were Carpet-bag politicians from the North and Republican-voting blacks. It carried on guerilla warfare for several years until the political accommodation of 1879.
The presence of social bandits such as Jesse James demonstrates sectional hostility. Immediately after the war, there was much banditry in Missouri; when these bandits were caught they were hung. But Jesse James advertised himself as carrying on against the bankers and railroads who were the beneficiaries of the northern victory in the 1861 War (and the railroads were one of the reasons taxes had been raised: railroad executives convinced the government to give them money). The agricultural people of Missouri (former Confederates) instead of hanging Jesse James, protected him from the law.
After the Civil War, sectional hostility abated relatively quickly (if we are talking about hostility that reached the level of war -- there still is some residual hostility) because there was no longer any real cause for the hostility. The war had been caused largely by sectional differences in culture and economy that revolved around slavery. Once slavery was gone as an issue, the tensions abated.
The North and the South had many ties. They were, after all, part of the same heritage of revolution and immigration. They were all Americans. Once the issue of slavery was removed, the things they had in common greatly outweighed the things that kept them apart.
The sectional hositility also abated quickly because of the North's lack of interest in forcing black equality. If the North had continued to impose Reconstruction in an attempt to help blacks achieve true equality, the hostility would surely have lasted longer.