First of all, let's explain what IMHO is the most probable way for Douglas to win.
It starts with Lincoln doing a little worse - just enough to lose Illinois, Indiana and California (to Douglas, duh) and Oregon (to Breckinridge).
Together with the OTL New Jersey division, that makes it 149 EV for Lincoln (three short of a majority), 73 for Breckinridge, 39 for Bell and - surprise - 40 for Douglas.

Now we have the Congress choosing between Lincoln, Breckinridge and Douglas. And of the 33 states, 15 northerns basically have a choice of Lincoln and Douglas, 15 southerns of Breckinridge and Douglas and the rest (Illinois, California and Oregon) unlikely to vote for anyone but Douglas.
Yeah, on the first ballot it's 15-15-3. But after a few dozen ballots, President Douglas.

Now, with such a result, the entire Douglas term (however much of it there is - OTL he died in 1861, but that might well not happen) would be full of "stolen election" cries
much stronger than
either 2000 or 1876.

What would he do with it? No idea.
...So what, how?


January First-of-May