Due to the rapid spread of antibiotic resistance, and the difficulties of treating biofilm-associated infections, alternative treatments for S. aureus infections are urgently needed. We tested the lytic activity of several wild type phages against a panel of 110 S. aureus strains (MRSA/MSSA) composed to reflect the prevalence of S. aureus clonal complexes in human infections. The plaquing host ranges (PHR) of the wild type phages were in the range of 51% to 60%. We also measured what we called the kinetic host range (KHR), i.e., the percentage of strains for which growth in suspension was suppressed for 24 h. The KHR of the wild type phages ranged from 2% to 49%, substantially lower than the PHRs. To improve the KHR and other key pharmaceutical properties, we bred the phages by mixing and propagating cocktails on a subset of S. aureus strains. These bred phages, which we termed evolution-squared (ε2) phages, have broader KHRs up to 64% and increased virulence compared to the ancestors. The ε2-phages with the broadest KHR have genomes intercrossed from up to three different ancestors. We composed a cocktail of three ε2-phages with an overall KHR of 92% and PHR of 96% on 110 S. aureus strains and called it PM-399. PM-399 has a lower propensity to resistance formation than the standard of care antibiotics vancomycin, rifampicin, or their combination, and no resistance was observed in laboratory settings (detection limit: 1 cell in 1011). In summary, ε2-phages and, in particular PM-399, are promising candidates for an alternative treatment of S. aureus infections.