(The full article is at the link if you scroll down a bit.)
One approach to this problem is to look to history (where instances of genuine moral progress are clearer) and see what methods of moral reasoning were used and to what effect. If we find methods that were reliably correlated with moral progress in that past, that may offer hope for the future.
Moral Bias and Corrective Practices: A Pragmatist Perspective looks at moral reasoning and its effects in the case of American slavery. It argues that "the moral biases of slavery advocates proved largely immune to correction by the dominant methods of moral philosophy, which were deployed by white abolitionists. Ascent to the a priori led to abstract moral principles—the Golden Rule, the equality of humans before God—that settled nothing because their application to this world was contested. Table-turning exercises were ineffective for similar reasons. Reflective equilibrium did not clearly favor the abolitionists, given authoritarian, Biblical, and racist premises shared by white abolitionists and slavery advocates."