The really interesting thing is that there are phages that have a broad host range (multi-species or even multi-genera), but due to the nature of how phages are typically isolated they are not often found by researchers. Phages, like other living organisms, show differences in ecological adaption with some being specialists (leopards) while others are generalists (rats). The specialists grow faster in their specific hosts than the generalists do so in the normal isolation process which uses a single host bacteria strain (plaque plate assay) and so they appear first. The generalists grow slightly slower and so are typically missed by the scientists.
In my lab we developed a different assay that favoured the isolation of generalist phages and we were really successful in finding broad host range phages (many were multi-genera in host range). I no longer work in this area and it is something I miss as it is such an interesting topic :(