Ah a question I can sort of answer - while not my chosen focus on the American Far Right I did study this area. Before answering it is important to remember that the KKK is a complicated beast. The area that you refer to is seen as the “Second Reconstruction” of the Klan where the KKK was reinvented through a mixture of factors including (but not limited to) patriotism, the film Birth of a Nation and I personally would argue fraternalism.

While some historians such as Chalmers view trends in the KKK linked to wars, and without a doubt the return of disaffected veterans brought a wave of men who could join the KKK, the Reconstruction Klan was very different in ways to the KKK of today. Miguel Hernandez argues that the Reconstruction Klan was essentially supported through the culture of fraternalism that just doesn’t exist anymore. During the 1910’s and 1920’s fraternities (basically clubs for men) were huge - the masons themselves had millions of members which contributed to the rapid spread of the Klan during this period.

William Simmons, founder of the Reconstruction Klan took huge inspiration from Birth of a Nation but failed to the really capitalise on the return of disaffected veterans following WW1. Rather, the revival of the Klan can be attributed to the strong xenophobic feelings at the time (particularly against the growing Catholic population) and the deeply successful recruitment strategies implemented by Taylor and Clark.

A quick overview of this recruitment by Hernandez demonstrates how the Klan often recruited key leading figures in towns or in other fraternities such as the Masons, leading to its spread throughout America. These factors can be seen as far more significant that any post war influx of disaffected veterans.

Most importantly it should be noted that the period following WW1 was largely one of prosperity. Therefore the Klan (while acting as a conservative force) was not one that truly attracted disaffected types like one might see today. This change occurred much further along the line nearer the fourth wave (1980s area) - an area I can expand on in far more detail in another answer but led academically by Kathleen Belew’s ‘Bring the War Home’ book.

To conclude briefly, the role of veterans in the Reconstruction Klan is minimal, particularly when compared to the wide scale association of disenfranchised veterans with the far right today.

A basic (if slightly problematic) overview of the Klan can be seen in Craig Wade ‘Fiery Cross’

For more era specific books search for Miguel Hernandez ‘Fighting Fraternities’ as well as Katheleen Blee’s ‘Women of the Klan’