IMHO TTL one aspect of Reconstruction that will have teeth in it will be the legal and political enfranchisement of the African-American population. Because of the greater use of black troops in combat roles TTL and also the more visible atrocities committed by the CSA in this regard, the Union will not allow the gains to be as temporary as they were OTL where within 10 years blacks were disenfranchised. There are several ways this can happen: more Union troops for longer in the former CSA to enforce "equality", stronger laws passed to enshrine black enfranchisement (and Supreme Court Justices appointed over the next 10-15 years who will make sure these laws stick - no slide to Plessey v. Ferguson), and disenfranchisement of CSA leadership both political and military - such as anyone who had a political office under the CSA government or a military commission is permanently barred from voting or holding any political or appointive office. With stronger laws, stronger enforcement, and sidelining of the wealthiest and most politically savvy elements in the south for a generation Reconstruction can be made to stick. It won't bring social acceptance or equality right away (though sooner than OTL), but it will prevent the abuses of the post Reconstruction which condemned African-Americans to segregation with poverty, political impotence, and limited educational opportunities.
Of course, the expected trials you'll see TTL will be a warning. Also, I can see the federal government strangling the KKK in its cradle, and N.B. Forrest ending up on the end of a rope for founding it (assuming he does not pull a "Ft Pillow" TTL which gets him hung at the end of the war).