You can't get a self-described Socialist Pope with any PoD after the publication of Rerum Novarum, the encyclical that outlines most of Catholic economic/social teaching. This encyclical, to give a TL;DR, condemns Marxist socialism on the basis of "private property is an inherent natural right," but also defends the right of workers to organize and calls for a living wage. After that encyclical is published, you cannot get any self-described socialist promoted to the ranks of bishop, let alone the College of Cardinals.
OP's exact words, indeed, already include the OTL Popes.
Literally Leo XIII.
But let's maybe be a bit more ambitious. Is it possible to have a Pope who favors revolutionary activity to implement such a "third way" economic system? Unfortunately, this is also tough, because the Papacy for much of the nineteenth century favored reactionary monarchies as a way of shielding itself from the possibility of bourgeois revolution. So you'd need a Pope who sees no future in cooperation with established monarchies and agitates for uprisings against them to create a just and Catholic social order--something kind of like theocratic Iran.
Something like fascism, but much more overtly Catholic, might also fit the bill. If Mussolini were a devout Catholic, perhaps...