Being "against Reconstruction" necessarily included opposition to black folks and their civil and political rights.

"Reconstruction" involved emancipation of the former slaves, but it also involved disenfranchisement of Confedrate veterans, which excluded the vast majority of white voters in the southern states from participation in the electoral process. Little wonder that so many newly freed black men were suddenly elected to many state legislatures, etc., and little wonder that so many white folks were enraged to the point of lawlessness.

Not to excuse the entire agenda of the KKK, but everyone should understand why southern white folks became so embittered. The gloating and punitive attitude of the victorious Union against the South worked against any reconciliation ,and encouraged racial and sectional hatred. President Lincoln, incidentally, strongly stated his intention to welcome the South back into the Union as graciously as possible, but his assassination put an end to any such efforts.