To pull this off you'd need to have a more organised left in the United States preceding the Great Depression. The US labor movement was very strong, but it was dominated by syndicalist tendencies that rejected electoral representation and focused exclusively on workplaces. This is what makes the US unique, I don't think the left actually 'was' weak, it just had a very different approach and focus than the European left.
The Progressive Party of the 1890s was huge, gaining nearly 10% of the presidential vote nationwide in 1892, even carrying a few states, and controlling the legislative of numerous states, however, it was slowly absorbed into the Democratic Party and therefore lost its appeal. The Socialist Party of America has potential to be a mass party. It controlled numerous local councils, and had numerous state representatives and a few congressmen. The trick I think for the US left to grow is if it follows the approach of the Minnesota Farmer-Labor Party, which started out as the Nonpartisan League and managed to gain a majority in the state legislative and elect numerous governors, including Floyd B Olson, throughout the 1930s.
So, lets say the progressive and socialist movement continues to grow, using entryist tactics into both the Republicans and Democrats to gain a foothold. Then, when the Great Depression hits, they win a landslide of the vote, which leads to a civil war from angry businessmen, opposed to their radical program of reform.