Nearly all of the "portrait" I'm about to paint draws from Eric Foner's oustanding Reconstruction: America's Unfinished Revolution, 1863-1877, though some of it is also tempered by WEB DuBois' Black Reconstruction in America.

Your question pertains primarily to the "first KKK", which developed into a clandestine cell (or splinter cell) group of anti-Union, pro-Confederate insurgents. This group was founded by a handful of former Confederate soldiers whose primary goal was to continue the defense of the Confederacy through guerrilla strategies. So, in a sense, the first KKK did conceive of themselves as restoring order and law to a land governed by lawlessness. It's just that the order and law the KKK sought to restore was the order and law of the Confederacy. (It should be noted that Gene Barber also wants to restore the order and law of the Confederacy.)

"Lawlessness" was absolutely common in the south following the Civil War insofar as the Confederate government ceased to exist following the Civil War. As such, there was a sort of power vacuum in the South which needed to be filled with new lawmakers, laws, and law enforcement. This was the motivation between Presidential (first Lincoln, then Johnson) and Congressional Reconstruction.

Now, the "general population" of the South was a lot more complicated than simply white and black. A small minority of Confederate symphatizers who were not active members in a pro-Confederate/White Supremacist group did approve of the plight and plan of the KKK (and other pro-Confederate/White Supremacist groups such as the Knights of White Camelia and the White League). According to Foner, most of the non-participant supporters of these types of groups were former members of the Southern Aristocracy or Slaveocracy. That is, most of the non-participant supporters were those Southerners who benefitted most from keeping things the way they were in regards to owning slaves. However, according to both Foner and DuBois, the majority of folks who were not former members of the Slaveocracy (i.e. working and lower-middle-class whites), were either totally ambivalent about or critical of the actions of the KKK (& other groups).

Scalawags (whites who supported the Union, but were from the South) were actively critical of the KKK & other groups (I'll get into more detail about why a lot of scalawags were so active in their criticism of the KKK down below). As such, they were targets of the groups' terroristic attacks nearly as often as were freed blacks. The logic was that if you killed one scalawag, you might dissuade someone else from ever "turning coat."

Carpetbaggers (northern whites who moved South, either to help with or profit from Reconstruction), also actively criticized the KKK & other groups. However, these groups tended to target carpetbagger properties more than the carpetbaggers themselves. The sort of dark logic was that if you killed one carpetbagger, another would move in to take his place. But if you destroyed the carpetbagger's home or office, then maybe future carpetbaggers would think twice before investing more resources in moving South.

Freed blacks, for obvious reasons, hated the KKK & other similar groups.

Interestingly, Foner notes that as the KKK stepped up their attacks, the Federal government also stepped up their Reconstruction efforts. In no other arena was this more obvious than in the Reconstruction efforts to build and develop a system of schools for newly freed blacks.

Throughout slavery, black slaves were denied the right to learn how to read and write. There were a series of laws in various states in the 1830s that made it illegal for anyone, white or black, to teach black slaves. According to Heather Andrea Williams in Self Taught, these laws followed the 1830 publication of David Walker's Appeal which saw the education of slaves as a stepping stone toward slave revolution, and Nat Turner's 1831 uprising, which Turner was able to organize and lead because of his high degree of literacy.

As such, following Appomattox, one of the first things that newly freed blacks sought was education. And one of the earliest moves in Presidential Reconstruction was the establishment of the Freedman's Bureau, whose primary responsibility was the establishment of a school system for newly freed blacks.

The KKK, still reeling from the Confederacy's loss, saw the education of freed blacks as the ultimate injustice. This was not only because there had been a generation-long assumption that educating blacks would give blacks the ability to rise up and overthrow whites, but also because up until that point in the South, education was a rare commodity. Indeed, in The Education of Blacks in the South James Anderson notes that prior to the system of schooling established (in part) by the Freedman's Bureau, the South knew no common, centralized system of schooling for whites or blacks.

And while the KKK saw the developing education system as a dagger in the belly of white supremacists, the vast majority of poor, uneducated whites in the South saw the developing education system as an exciting new possibility. That's because the Freedman's Bureau was essentially establishing an education system for all Southerners, not just black Southerners. As Anderson notes, with the development of education for blacks Southerners came the development of universal education for all Southerners.

As such, when the KKK bombed Freedman's Bureau offices, or when they burned down newly built schools for freed blacks, many poor whites got pissed because the Slaveocracy wasn't, they realized, beneficial to all whites; it was only beneficial to wealthy whites. Consequently, one bizarre consequence of the KKK attempting to restore Confederate law and order to the South was that they actually created people who were opposed to that order.