As someone who is interested in the idea of a strong American Socialist tradition there are a number of PoDs I can think of that could go some way towards establishing the Socialist Party, or at least a socialist Party, as a major force in American politics:
Wanking their successes in local and state elections.
@SaltGiver has a number of threads that are a veritable treasure trove of elections that socialists came close to winning at various times. Giving them a stronger presence throughout the country could help them to build their profile and reputation as a credible alternative, whether that alternative is a radical repudiation of the wage system or sewer socialists who present themselves as better administrators than the mainstream parties.
Have
Max S. Hayes succeed in replacing Gompers as President of the AFL in 1912. Admittedly a pretty tall order, given that he only got about 30% of the vote IOTL and Gompers' position was pretty well entrenched. One idea I had was
having the Supreme Court not overturn Gompers' conviction in a year earlier in 1911 which could possibly result in a more divided leadership election with Hayes somehow coming out on top. Having a committed socialist in charge of the AFL could go some way towards strengthening and legitimising the Socialist Party. Removing Gompers could also lead to the AFL being more open to
Industrial Unionism, which I see as more conducive to class-based politics and was generally supported by American socialists at the time.
Keeping the US out of WW1. WW1 was a disaster for the American socialist movement. First the war split them between the nationalists and pacifists. Then the Russian Revolution split them between reformists and revolutionaries. Finally the Red Scare killed off their growing momentum and they never really recovered from it. Keeping the US neutral, whilst it may not prevent the Russian Revolution or even Red Scare entirely, could put the Socialists in a stronger position.
Have the
Conference for Progressive Political Action agree to form a united party, as was proposed in 1922. Doing so would not only allow the Socialists to unite the disparate progressive factions in US politics under their banner, but it could also mean that said party would end up taking La Follette's place in the 1924 election, assuming that the Democrats and Republicans still both nominate conservative candidates, given that La Follette was fundamentally still a Republican who wanted to prove that there was still demand for Progressive Republican politics. If they have a result even remotely as good as La Follette had IOTL it would be a good way to establish them as a force in US electoral politics on a national level. Furthermore, if they can retain their relevance for another 8 or so years they will be in a good place to capitalise on the Great Depression, assuming that still goes ahead.