The obvious slot for a third party in the US

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A Black Christian party.

The place it would contest for Congress are black majority-minority districts, in particular the ones so uncompetitive currently that the Republicans don't run anyone.  Politically, that works especially well because in an unconstested district, one doesn't have to worrry about those who favor your party making a lesser-of-two-evils calculation and voting for the "less evil" major party candidate.  it's this that makes such a party more viable than anywhere else; "solidly" Republican districts have 2-to-1 majorities, not the 4-to-1 of some of these places.  And they're largely immune from being gerrymandered out of existence, given the current Voting Rights Act interpretations.

The retail political courting would logically initially center on pastors and congregations of black churches, perhaps the most important non-givernment organizations in the districts in question.

The platform?  It's easy to make a list of Democratic policy positions rejected by the polled black voter.  Supporting school choice, school prayer, and funding for faith-based initiatives while opposing gay marriage and low-skill immigration, for example.

Could it go contesting nationally?  Perhaps not.  In states where candidates can have multiple party endorsements (for example, New York), it could nontheless be a significant factor in state and local races.  And as merely a party represented in Congress and state legislatures, it could weild significant influence by its ability to ally alternately with Republicans and Democrats.

The remaining factor then is how long it would take for a major party to adjust such that it could absorb the voters from a Black Christian party.  De facto, the Democrats already have (insofar as one hasn't already emerged), so they'd probably be in the best shape.  However, competing against such a party would be more difficult; much of the party's progressive base would find accomodation on the issues the Black Christian Party would use to win votes very distasteful.  It would seem likely that they would depart for Green-er pastures.  The Republicans would have similar fracturing occuring to their coalition.

The most interesting outcome possibly would be another major realignment.  Perhaps, on one side, a party of Christian values, skeptical of free trade and of immigration, favoring a social welfare state administed by faith-based organizations.  On the other, a libertarian, areligious, free trade, balanced budgets, low tax, pro-business party.  Progressives would be split between them, much like modern libertarians are split between the two current parties; the Green Party would be a marginal party filling the role filled for so long by the Libertarians.