If "by that time" you mean Reconstruction and the post 14th & 15th Amendment era, probably the majority of many southern states were Republican. That would include virtually all freed slaves and a sizable percent of white Unionists who never supported the Confederacy. Georgia, Alabama, Mississippi, Florida and other deep-south states that had majority or near majority black populations and also had Unionist sentiments sent Republicans to congress and elected Republican majorities to state legislatures, etc. That is why the KKK was founded --- to keep blacks away from the polls and return Democrats to office. The function of the KKK and related groups from the late 1860s until the 1960s was as the enforcement arm of the Democrat party, and they did their job with exceptional, if brutal efficiency.
Even today, you will still hear of elections in southern states that mark "the first Republican since Reconstruction or some such wording. (I think the current Texas legislature is the first Republican majority since Reconstruction.) If the current crop of race hustlers really wanted to sue an institution for support of slavery and civil rights abuses, they should sue the Democrat Party instead of some Insurance company or railroad that had some remote tie at best to past slavery and discrimination. The Democrats have a documented, totally non-interrupted 160 year history of championing both slavery and racial discrimination.
To Joe Stalin
The Republican Party was formed around 1858, out of the ashes of the Whig Party. The Whigs, were pro national bank, pro tariff, and in favor of supremacy of the Federal government over the States. The Republicans, garnered most of the Whigs, and the abolitionists as their base
Fractured history in the flavor of your namesake. The Republican Party was formed in 1852 in the Midwest as a Free-Soil party who's platform was entirely opposition to the expansion of slavery to the West. They were not necessarily abolitionists although some abolitionists (not all) were attracted to their platform as more politically realistic in the pre-Civil War days. (just as many Libertarians register and vote Republican today -- a more realistic vote as opposed to the Lou Rockwell no-chance party)
Whigs could come down either way on slavery with the general divide being that Northern Whigs were likely anti-slavery and anti-expansion while southern Whigs were pro-slavery and pro-expansion. That divide is what killed the Whig Party in the 1850s with the Kansas-Nebraska Act and the Fugitive Slave Act. While the Whigs were divided on slavery, the Democrats were united. There was no such thing, North or South, as an anti-slavery Democrat! As the Whigs collapsed, Southern Whigs like CSA V.P. Alex Stephens became Democrats while Northern Whigs like Lincoln became Republicans. But there was no such thing as an anti-slavery democrat. North or South, they all actively supported it or were not opposed to it and would vote for those who did support it.