Antagonistic co-evolution between hosts and parasites (reciprocal selection for resistance and infectivity) is hypothesized to play an important role in host range expansion by selecting for novel infectivity alleles, but tests are lacking. Here, we determine whether experimental co-evolution between a bacterium (Pseudomonas fluorescens SBW25) and a phage (SBW25F2) affects interstrain host range: the ability to infect different strains of P. fluorescens other than SBW25. We identified and tested a genetically and phenotypically diverse suite of co-evolved phage variants of SBW25F2 against both sympatric and allopatric co-evolving hosts (P. fluorescens SBW25) and a large set of other P. fluorescens strains. Although all co-evolved phage had a greater host range than the ancestral phage and could differentially infect co-evolved variants of P. fluorescens SBW25, none could infect any of the alternative P. fluorescens strains. Thus, parasite generalism at one genetic scale does not appear to affect generalism at other scales, suggesting fundamental genetic constraints on parasite adaptation for this virus.