Although this is an unlikely POD, the late 1860s/early 1870s would be an interesting time for a socialist movement.
Most likely it would not be "socialist" as we recognise it. Some of the most important early voices in American socialism, such as Daniel De Leon, had not really established themselves by this point. Nor had Henry George's ideas galvanised the "progressive" or "populist" movement as it was termed.
Much like the Paris Commune of 1871, this "socialism" would be much less Marxist (who only really achieved his political ascendancy post-1871 in part by writing the post-script on the Commune) and much more anarchist in nature. Although again it might not be referred to, at least by supporters, as anarchist. I expect it would have utopian tinges and have an overall co-operative and Proudhonian flavour. It would be less likely to attract black support, or court it, depending on where in the country it occurred. Remember by no means did a majority of Northerners in the 1860s believe African Americans could be social equals. Women - harder to say. Many labour activists resisted calls for the emancipation of women because they felt that the ideal working-class home was where a man could bring home enough money to support his wife and children without them having to work. Depends, again, on where this emerges and who supports it.
You might want to look at movements like Chartism in the UK, active in the 1830s and 1840s, to give you a sense of what this pre-socialist radicalism looked like. Or perhaps take some inspiration from the LocoFocos and Young America movements.