Kronecker coefficients encode the tensor products of complex irreducible representations of symmetric groups. Their stability properties have been considered recently by several authors (Vallejo, Pak and Panova, Stembridge). In previous works we described a geometric method, based on Schur-Weyl duality, that allows to produce huge series of instances of this phenomenon. In this note we show how to go beyond these so-called additive triples. We show that the set of stable triples defines a union of faces of the moment polytope. Moreover these faces may have different dimensions, and many of them have codimension one.