What did southerners referred to their resumption of power in the region as?

The answer is redemption. In United States history, the Redeemers were a political alliance in the Southern United States during the Reconstruction Era that trailed the Civil War. Redeemers were the southern wing of the Bourbon Democrats, the traditional, pro-business faction in the Democratic Party, who shadowed a policy of Redemption, seeking to exile the Radical Republicans, a coalition of freedmen, "carpetbaggers", and "scalawags". They normally were led by the rich landowners, businessmen and professionals, and dominated Southern politics in most parts from the 1870s to 1910.