I'll preface this by saying that I know very little about the American Civil War and might have the complete wrong idea, so sorry about that. I'm also aware that the Confederacy can be a touchy subject and some people choose to try and adapt history to fit their view on the subject. I'm from the UK so have no opinion either way, I'm just trying to better understand it!
Now, as I understand it, many areas fought for the Confederacy that really had very little to gain from slavery due to the terrain and scale of farming being employed meaning that the use of slaves really wouldn't have added anything to their economy. I believe that Tennessee had many such areas, while other parts of the state benefited more strongly from it.
To take that example, as it seems a tragic one, how did they end up siding with the Confederacy, especially with the large amount of destruction that happened on their land? How were they able to not only drum up enough support from those with nothing to gain from it, but also keep them on side while it was so heavily damaged by the fighting there?
It seems that the support for the Confederacy should have been tenuous at best, and the danger of counter-rebellion along with the destruction they faced must have been seen as an unsustainable option, or at least would have considered surrender when things started to stack against them.
You pose a good question, and surprisingly, what is needed to get at the heart of it is a modern perspective. More on that in just a minute, though. First, a little background.
So, the states that seceded from the Union did so individually by votes on the state level. When a state like Tennessee seceded, it then became eligible for membership in the new Confederate States of America, which organized in Alabama to form an official government with a constitution and elected officials (this government later relocated to Richmond, Virginia). These officials elected Jefferson Davis to prosecute the war for them as president. Although the individual states sent regiments off to fight, and this organization was done on a very local level, the states themselves had very little say in how those regiments would be used. To go even deeper, the soldiers themselves obviously had even less to say about how they would be used in the war, and by extension, their families and friends (unless elected officials in the CSA government) had even less to say about things. So in this way, the "people" really didn't have a lot to say about whether their state seceded, or afterwards, whether they would stay that way. Their opinion mattered, to be sure, but it was the politicians who pulled the trigger on disunion, and they did this for reasons directly tied to slavery, yes, but also because of a long-standing rift between north and south and ideas about big vs. small government.
I only bring this up to illustrate how organized the Confederacy was in terms of its military and legislative command structure. This was not a rebellion of hastily organized malcontents, but rather the result of a number of very experienced politicians who had been scratching and clawing over the slavery question, and even before that, the big government vs small government question, for decades. These southern politicians perceived Lincoln's election as a signal that the abolition movement was going to spread south of the Mason Dixon line, and they acted according to that perceived threat.
Thus, to answer your question, the reason that so many people (and regions) in the south fought for the Confederacy despite the fact that the slavery question didn't really affect them is because those people didn't necessarily have a say in whether or not their state seceded. Their civic leaders made that decision for them, and once made, it was often a matter of regional pride (one might even say nationalism) that drove individuals to enlist. The southern narrative was changed on a micro level to "state's rights" and defense against the "invaders," which is a myth that still pervades those regions to this day, since it is hard to rally around a cause whose foundation is cruel despotism.
In reality, the slavery question was the overriding issue that led to civil war, and it was that issue that drove the civic leaders of the individual states. Still, it is important to remember that slavery wasn't the only cause, and that the American Civil War actually had roots in the big (Hamilton) vs small (Jefferson) government debate the founding fathers had almost immediately after the revolution.
Again, back to a modern perspective. Much like today, it wasn't necessarily the people who decided to go to war, but rather the elected officials, who were drawing on both a hot-button topic, but also decades of unresolved conflicts rooted in different ideas about the role of the government in state affairs. Similarly, in a modern war, it isn't like the people vote on whether or not to go to war, elected officials do that. It was very much the same in the US Civil War.
[Sources - James McPherson, 'Battle Cry of Freedom'; Shelby Foote, 'The Civil War: A Narrative, vol. 1'; Bruce Catton, 'Mr. Lincoln's Army']
In Arkansas, specifically, the northern part of the state is mountainous and there were few slaves in this area, but they did exist. Anyways, The first secession convention rejected the ordinance for secession for the first vote. This was mostly do to the fact that most of the delegates were pro-union or conditional unionists. After the firing on Ft. Sumter and President Lincoln's call for troops to end the rebellion, Arkansas's Governor Rector (who was an ardent secessionist), recalled the convention back into secession to re-vote. The second vote passed, with only 5 voting against the ordinance of secession (Mostly due to the fact that the conditional Unionists switched over to the secessionists, because they felt that Arkansas was going to be used as a road to invade their neighbors who already seceded.) The Convention president urged the 5 who voted against to reconsider their vote by saying (basically) "If we must go, let us show the country that Arkansas is united." 4 of the 5 changed their vote. Most of these men changed their vote, because if they were going to war, they were going to help defend those who they shared a common life, history, culture and ideals.