Emerging classes of concentrator photovoltaic (CPV) modules reach efficiencies that are far greater than those of even the highest performance flat-plate PV technologies, with architectures that have the potential to provide the lowest cost of energy in locations with high direct normal irradiance (DNI). A disadvantage is their inability to effectively use diffuse sunlight, thereby constraining widespread geographic deployment and limiting performance even under the most favorable DNI conditions. This study introduces a module design that integrates capabilities in flat-plate PV directly with the most sophisticated CPV technologies, for capture of both direct and diffuse sunlight, thereby achieving efficiency in PV conversion of the global solar radiation. Specific examples of this scheme exploit commodity silicon (Si) cells integrated with two different CPV module designs, where they capture light that is not efficiently directed by the concentrator optics onto large-scale arrays of miniature multijunction (MJ) solar cells that use advanced III-V semiconductor technologies. In this CPV⁺ scheme ("+" denotes the addition of diffuse collector), the Si and MJ cells operate independently on indirect and direct solar radiation, respectively. On-sun experimental studies of CPV⁺ modules at latitudes of 35.9886° N (Durham, NC), 40.1125° N (Bondville, IL), and 38.9072° N (Washington, DC) show improvements in absolute module efficiencies of between 1.02% and 8.45% over values obtained using otherwise similar CPV modules, depending on weather conditions. These concepts have the potential to expand the geographic reach and improve the cost-effectiveness of the highest efficiency forms of PV power generation.