Children from low-income families often enter kindergarten academically behind their higher income peers. Recent findings also indicate gaps in social-emotional aspects of school readiness, raising questions about cross-domain relations. Using a subsample (N = 3,485) of 3- and 4-year-olds attending center-based programming in the Head Start Impact Study, we investigate whether improvements in children's social competence may mediate the effect of Head Start assignment on children's early math skills. We posit that social competence can be effectively captured via a combination of two distinct constructs: social skills when interacting with peers and teacher-child relationships. Indeed, confirmatory factor analyses revealed good fit for a measurement model of social competence using indicators of these constructs, each of which measure different, yet complementary, aspects of children's interpersonal skills. Structural equation modeling showed that, although random assignment to Head Start did not impact early math skills or social competence in this subsample, children's social competence was positively related to math achievement during the preschool year. Taken together, results suggest that preschoolers' relationships with others -- effectively interacting with both peers and teachers in classroom settings -- positively predicts math skills. Findings can guide curricular decision-making and time allocation, particularly in preschools serving children from low-income families.