Highly effective HIV-1-neutralizing antibodies could have utility in the prevention or treatment of HIV-1 infection. To improve the potency of 10E8, an antibody capable of near pan-HIV-1 neutralization, we engineered 10E8-surface mutants and screened for improved neutralization. Variants with the largest functional enhancements involved the addition of hydrophobic or positively charged residues, which were positioned to interact with viral membrane lipids or viral glycan-sialic acids, respectively. In both cases, the site of improvement was spatially separated from the region of antibody mediating molecular contact with the protein component of the antigen, thereby improving peripheral semi-specific interactions while maintaining unmodified dominant contacts responsible for broad recognition. The optimized 10E8 antibody, with mutations to phenylalanine and arginine, retained the extraordinary breadth of 10E8 but with ∼10-fold increased potency. We propose surface-matrix screening as a general method to improve antibodies, with improved semi-specific interactions between antibody and antigen enabling increased potency without compromising breadth.