Trouble is, it's hard to see any scandal being big enough to change the outcome in 1920. The Democrats were humungously unpopular [1], not just losing the Presidency, but being all but annihilated in Congress outside the South.
If one blows up during the campaign, Harding can probably brazen it out and deny everything - after all, nothing to his dicredit was known as at the time of the Republican Convention. If he can't, then probably the Republicans hastily pick another candidate - probably Coolidge. Any story that explodes too near the election for that is almost certainly too late to prevent Harding's election.
For Cox to win, you need an earlier PoD. If Hughes wins in 1916, and takes America into WW1 as Wilson did, then the backlash against wartime sacrifices will fall on the Republicans instead of the Democrats. If health issues prevent ex-President Wilson making a comeback, then Cox is as likely as anyione.
[1] NYT columnist Charles Willis Thompson remarked that in 1920 the Republicans didn't really need to nominate anyone at all. The Democrats were so unpopular that the people would simply have voted for Republican electors, and let them choose whoever they wanted as President.