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As a next step in the exploration of a class of flavored gauge mediation models of supersymmetry breaking in which the mixing of the Higgs and messenger doublets are connected by a discrete non-Abelian symmetry, we investigate the generation of nontrivial mass hierarchies and mixing angles for the Standard Model matter fields. We consider here the case in which the Higgs-messenger symmetry, which we take for concreteness to be the discrete group $$\mathcal{S}_3$$, also plays the role of a (partial) family symmetry. Within the specific implementation of $$\mathcal{S}_3$$ studied here, we show that couplings at the renormalizable level can resultmore » in hierarchical quark masses, but do not lead to phenomenologically viable quark mixing parameters, requiring the inclusion of higher-dimensional operators. As a concrete exploration of this idea, we show that the Cabibbo angle can be generated within this framework via such nonrenormalizable couplings and explore the phenomenological implications of this scenario.« less