We had a few good third party attempts: Populists, Roosevelt, the Dixiecrats, Wallace, and Perot, but they weren't the four way messes 1824 and 1860 were.
How can we get another one of those elections, with at least 4 plausible candidate winning decent amounts of both electoral and popular votes, with the possibility of a tie?
The Democrats nominate a conservative in 1932--or they get one through FDR being killed. In 1936, the conservative Democrat (maybe Garner) manages to get re-nominated through his control of the party machinery, but a large left-wing contingent leaves the Democrats. So there is a four-way race in 1936:
(1) The Democratic Party--rather conservative, for states rights, a low tariff, some antitrust, maybe some limited intervention to help farmers. About all it has to offer labor is restrictions on labor injunctions by the federal courts. Its big strength is in the Solid South, also with traditional Democrats in other parts of the country.
(2) The Republican Party--conservative, business-oriented, though with a mildly "progressive" minority who stay with the party partly from inertia, partly because they view the party as the symbol of respectability--they associate the Democrats with the South and unsavory urban machines (and fear the Farmer Labor Party as too radical).
(3) Farmer-Labor Party--This idea was "in the air", and there were state Farmer-Labor parties in Minnesota and elsewhere. What prevented the idea from succeeding nationally was that most of its potential supporters were co-opted by the New Deal--which doesn't happen in this TL. Supporters of the new party would include trade unionists, dissatisfied farmers, liberal intellectuals, socialists, and the Communist Party in its Popular Front stage. (If the national FLP were formed anytime between 1929 and 1934, the Communists would at first denounce it as "social fascist" until the 1935 Comintern congress would set them straight.) If the conservatives retained control of the Democratic Party in 1936, there could be a mass exodus of "progressive" Democrats to the new party.
The future of the Farmer-Labor Party would be uncertain of course--especially after 1939 and the Hitler-Stalin pact, the issue of Stalinist influence within such a party would be explosive, and there would be battles between Stalinists and anti-Stalinists in the party, similar to those which took place in OTL in the Minnesota Farmer Labor Party and New York's American Labor Party.
(4) A Social Justice party--anti-Marxist, anti-liberal, isolationist, populistic. Blames "international bankers" for the Depression. Called "fascist" by its opponents, but it denies it has anything in common with "foreign isms." Led by Huey Long (in this TL his assassination is butterflied away) and supported by Father Coughlin, Townsend, etc.