early on, the state parties might remain similar ideologically to OTL- that is having a agrarian populist party and a urban liberal party like it happened OTL. Overtime, the issue of slavery, industrialization, social strife and westward expansion would change the national politics tremendously. For instance, Oklahomma might preserve its socialist movement, which was strong in the late 19th and early 20th centuries while Appalachian states will certainly develop some sort of Labor Party- the role of the coal industry in the Appalachian economy is just too big for that not to happen. Kentucky, Tennessee and Alabama are likely candidates to have a Labor Party.
One of the problems of the socialists is that the two-party autocracy was too entrenched so if they have to focus on the regional, state politics the left will certainly thrive. It defenitely was doing so OTL before the SPA was crushed by Woodrow Wilson.