I gather the Democrats didn't get all their ballots organized. There was a Douglas slate of electors and a fusion slate of anti-Republican Douglas-Breckenridge-Bell electors.
Only the three Douglas men who appeared on both ballots beat the top four Republican electors. Those who were only on one of the two competing Democrat ballots lost to the Republicans. That is if I've read and understood this correctly.
It's certainly an interesting case, but I don't think it significantly changes the popular or the (all important) electoral vote count.
FWIW Something not so very different happened a century later which leads many to conclude that Nixon actually beat Kennedy in the popular vote. It's not about fraud in Illinois or Texas (though that matters as well).
Rather it's that two sets of competing Democrat electoral slates in Alabama were both counted as votes for Kennedy by the media. If you count the votes for the independent slate as not Kennedy's Nixon carried the national popular vote.
It’s the last time there’s been more than one faithless elector in any presidential election.
Can you imagine that 1860 election today?