(Almost aside: Why does everyone discard the intuitionists and constructivists? The history of this is that Hilbert, the Formalist challenged Intuitionism as an alternative to Platonism and Hilbert's program lost. It relied upon the completeness of arithmetic, which was formally disproved. But his approach gets to be the major contribution from the confrontation?)
Intuitionism and Fictionalism are two very interesting views of math that I think are coherentist at their core.
If, as proposed by the former, mathematics is an art form based on evolved suppositions, it just extends assumptions that are not true, only necessary for humans. All of our science is couched in it, not because it represents something real, but because it captures what we reliably understand.
Likewise, if as the latter suggests, Platonism is obviously false, but it is reliable as a limited playground for the comparison of possibilities, then again mathematics is held together entirely by language and shared imagination, not truth. This gets us the same answer without imposing a theory of the human mind.
In both cases, all you get is coherence, not grounding in reality or Formalism's sort of transcendental clarity (that is always perfect by virtue of never necessarily meaning anything.) And counter to the thread in the comments, nobody prevents you from including new intuitions or from positing random axioms just to see whether they become appealing. So neither of these approaches is limited to formal derivability. In fact, within early Intuitionism, Brouwer expressed great disdain for Heyting's formal derivations.
You can also adopt the framing of Intuitionism (of relativistic psychological Platonism, upholding Fictionalism) without fully adopting the fussy conservatism of its founding cadre. Famous Intutionists like Steven Kleene have done classical math... (But that fussiness does keep you aware that using concepts with known paradoxes in them, like absolute negation, must always be done provisionally.)
You can best characterize the actual behavior of most modern mathematicians as a faith in local Platonism but involving a limited pluralism that directly implies a complete Platonism is false. There are still bounds on the pluralism, and they are set by coherent overlaps between locally Platonic 'pictures', which they assume all hang together in the end. This is a logic consistent with Fictionalist formalizations, even if practitioners would find the overall framing abrasive.
One vision of this many-worlds-but-not-too-many approach is represented by the search for 'Ultimate-L', a map of all the relationships between possible set theories that are not too bizarre to use.
Both these ideas are from the first half of the previous century, so I don't know whether that is 'recent' in terms of the question. Intuitionism arose in the 1920's. Fictionalism is a way of elaborating on the theory of meaning that proceeds from Wittgenstein's approach that "meaning is usage", which is from some time in the mid 1940's (though this is confused by his reluctance to publish.)