I think it would probably take a treatise to answer this fully, but I have actually been looking into some of the source materials on this, so let me start with a bit from the last thread, that one of my main references is John Coski's The Confederate Battle Flag.

The "Confederate Flag" as we know it (blue saltire with thirteen white stars on a red field) began as the battle flag of the Army of Northern Virginia in 1860, and by 1863 was being incorporated into the Confederate Naval jack and ensign, and as part of the national flag of the Confederacy (replacing the "Stars and Bars") by the end of the war, it was effectively the popular symbol of the Confederate States of America.

That was 1865. I haven't found much about the next thirty years or so, because I haven't really looked for it, but by the 1890s you had states like Mississippi adopting it as part of the state flag, and the Confederate flag had been used by many groups like the United Confederate Veterans and United Daughters of the Confederacy as part of their imagery and Confederate memorials.

What about 2nd Ku Klux Klan or Birth of Nation - flag used?

The Confederate flag does feature in the 1915 film Birth of the Nation, which provided the seminal inspiration for the second embodiment of the Ku Klux Klan, which grew to 2 million members during the 1920s. However, the KKK during that period was a fraternal organization organized like a pyramid scheme, and their imagery was based more on the film than anything related to the CSA, and their particular biases were geared toward whatever the local issues (blacks, Jews, immigrants, etc.) were, rather than a consistent ideology; they often used the American flag, but not the Confederate flag. (Coski 84) The sordid murder trial of one of the klan's leaders in 1925 led to its rapid demise.

However, the image of the Confederate flag continued to be used as a symbol of Southern heritage, particularly in connection with groups like the UDC and in connection with Lost Cause literature throughout the 1920s and 30s, and hit the pop zeitgeist with the release of Gone With the Wind (1939). (Coski 90)

In this context, there were at least two important movements. One: the diminished Ku Klux Klan, still being led by Hiram Wesley Evans in the 1930s, was progressively more Southern, more racist, and more violent; Two: the Confederate flag was increasingly considered a positive symbol of Southern identity. The two movements sometimes conflated, in Fighting the Devil in Dixie by Wayne Greenhaw, he describes how KKK members in the 1950s prominently displayed the Confederate flag. In other instances, like the use of the Confederate flag was deliberately meant to have positive historical associations; the badge of the US Army Rangers for example formerly included a quartering of the Confederate flag, representative of tracing the heritage of ranger units in the US to John Singleton Mosby; and their were other military uses, though officials tried to curb US military usage of the flag in the 1950s. (Coski 115-118)

Taken together, the heavier usage of the Confederate flag after its popular revival in the 1940s and 50s, coupled with the increasing awareness of race relations, particularly in the Southern United States which the flag came to represent, and with the face of Confederate-flag-waving racists like the Ku Klux Klan (and, more generally, the States Rights Democratic Party or "Dixiecrats" created by Strom Thurmond in 1948), probably helped cement the image of the Confederate flag in at least some minds as a symbol of racism and white supremacy. The military, at least, was aware of this and took active steps against official displays of the flag in the 1960s, alongside other steps related to desegregation. (Coski 156)

Personally, I'm not sure what real difference there is between the push against the Confederate flag in the 1960s and the push against the Confederate flag in the 2010s, as in both cases there were battles to remove flags from government buildings, colleges, etc. To quote an important passage from Coski (157):

Historians cannot say whether the flag controversies of our own time would have occurred if the battle flag had not become an overt and intentional symbol of racism in the 1950s and 1960s. The preeminent symbol of our country's most significant dissident political movement is destined to be controversial, especially when that movement was so closely related to racial issues. If the symbol were merely a historical one, however, the passions surrounding it would not have been so strong or immediate. Instead, the Confederate flag is not simply a symbol of perceived racism in the country's distant past but a symbol of explicit racism in the recent past and in the present. Sometimes consciously, those who have used the Confederate flag as a racist symbol in recent decades have solidified the perceived link between modern racism and the Confederacy.

In part, I think it depends on where you stood. There were Southerners in the 1960s that decried the usurpation of the Confederate flag by racists like the Ku Klux Klan (Coski 158-159), and even as late as 1979, when the Dukes of Hazard went on air, there was evidence of the flag as a positive symbol of Southern identity. And just the same, even today we deal with the issues of racism and heritage associated with the flag.

So to sum up a little, the flag went from a battle standard to a sort of general symbol of the Confederacy as a whole (1860s-1870s), became co-opted into Lost Cause symbolism (1870s-1910s), became associated with broader regional, ideological, and political identities (1910-1940s), was gradually degraded in association with racist groups, views and organizations (1940s-1960s...technically still ongoing), and challenged (1950s-now).

There's a lot of aspects I haven't touched on - like I said, I'm primarily summarizing a lot of this from Coski's book, with a few supplemental sources - and personally I'm curious when the Confederate flag became the merchandising juggernaut that it is today; I strongly suspect it owes itself to the massive and enduring popularity of Gone With the Wind, but I haven't found any good sources on that yet.