I kept it as 1861 because maybe the civil war itself could go differently or be less conclusive (although the CSA already had a lot of factors working against them). My knowledge on this period is limited to reading from around the board.
I believe the reference is to the fact it should be no earlier than 1861 or later than 1914.
At any rate, Hayes was willing to agree to a lot of Tilden's sagenda and to remove Federal troops, so what I think you need is a bit earlier.
Let's say that President Grant's insistence on ending the Klan leads to his assassination after newspaper reports show he's sure to wint he election (and Greley dies anyway). The Radicals are enraged at the South for this, but now, like the Democrats, their electors are free to vote for anyone. Southern governors are enraged as Federal troops become violent agaisnt the Klan over the murder of Grant, and as V.P. Colfax, a real lame duck as President for a few months, tries to calm things supporters of him suggest he should get Grant's electoral votes isntead.
So, a divide int he GOP at a crucial time causes violence to break otu more tand more in the South, and while states might not secede, the idea of guerilla warfare - proposed by some near the end but quashed by men like Lee - gains traction. With Longstreet one of few opposing it being practicallypersona non grata in Southern circles for being a Republican (and for disagreeing with lee, who was lionized), a major problem erupts.
if the Panic of 1873 can come a bit earlier and throw things into more of a mess, you might be able to make it a 2nd Civil War and not just a small insurrection.