For reconstruction to succeed, the freed slaves and their white allies will need to be able support themselves economically and defend themselves militarily. The former will generally require land reform; if freed slaves actually got 40 acres and a mule, you wouldn't have sharecropping, radical state governments would have durable tax bases, and the planter oligarchs who funded the redeemers would have been expropriated. In theory this could be done in a way to divide poor whites from the planters, if they were ensured some land too. Taking away the funds the planters used to fund the superior firepower they used to overpower republican militia would help militarily, as would getting more loyal poor whites, but durable free militia would probably require greater changes.
Preventing Andrew Johnson would be a major help towards both those goals. Another way, that might allow for deeper changes, would be if a white insurgency developed immediately after the defeat of the Confederacy, rather than simmering for a few years. By the end of the war black troops made up a sizable and growing part of the Union army; going straight from Appomattox to counterinsurgency would allow the peak of abolitionist military strength to be used to weed out Klansmen, and would likely produce sturdier pro-union local forces.
Long term though you'd need to keep pro-reconstruction majorities in Washington for longer, and the best way to do that is to avert the Panic of 1873, which led to a Democratic electoral resurgence. Different silver policy could help there, although some of that depended on Europe. Less republican corruption could help, although that was always a bit overstated. More responsible railroad construction could avoid the wave of bankruptcies that sparked the panic. The failure of the Northern Pacific Railroad was a key trigger; prevent that, or have a federal rescue plan come in, and the whole economy would benefit. Federal intervention, in 1873 or earlier, could additionally help if it involved landgrants to freed slaves or black veterans. A large, federally assisted settler movement could bulk up the Northern Pacific's customer base, and a couple black senators from Dakota or Montana could have a big impact in Washington. More judicious use of black settlers could also lead to more peace with the Native Americans, by creating a firebreak against white encroachment into treaty lands. A bit more forced migration of unreconstructed confederates, either into the Southwest or abroad, might also ease the pressure on reconstruction governments.
A couple other potential changes that would help:
- Bringing Haiti and the DR into the US could provide important allies for reconstruction. The key here would be to bring them in as states, not territories. They almost joined IOTL, but this was scuppered by the radical republicans as they didn't think unrepresented territories full of black people would get fair treatment.
- Changing the Paris Commune to make international radicalism either more palatable to mainstream America, or less of an immediate threat. There were a fair amount of socialists behind reconstruction, and the horror in certain circles at the commune turned a lot of moderate northerners off radicalism while giving conservatives an additional, less racist, set of talking points.
- Women's suffrage! Early on in reconstruction there were a few state-level efforts that barely failed (Kansas and New York, iirc). Have them succeed, and/or have some of the reconstruction states give women the vote on principle, and it opens up lots of possibilities. White women could be brought over to reconstruction, and women generally would widen its base, especially as so many men died in the war. Lots of women were involved in the abolitionist movement, and debates over giving black men the vote first fractured that coalition. Give some women the vote, and make that a viable movement overall, and not only would the radicals be more united but some of their best leaders would be allowed greater prominence. This could also help undercut racist rape narratives, though it might also empower them in certain instances.
- Have Grant be the moderate candidate in 1872, not the radical. The outcome wouldn't change, but it'd be easier to have the Democratic Party stay dead, and would shift the overall center in a more durably pro-reconstruction direction.