An important factor in grid integration of solar power is the so-called dispersion-smoothing effect, i.e., that differences in cloudiness over dispersed systems make the aggregate output less variable. This effect has been studied for irradiance step-changes on different time horizons, but not so much for instantaneous irradiance. In this paper, an improved probabilistic model is proposed for how instantaneous solar irradiance is correlated and aggregated over a network with arbitrary number of sites and dispersion. The model is fitted to irradiance data with a 1-s resolution from a network with 17 pyranometers in Hawai'i. A previously proposed three-state model of the instantaneous clear-sky index is partly confirmed by showing that clear and cloudy states can be separated and modeled by independent distribution models. It is also shown that the station-pair correlations for the instantaneous clear-sky index, as well as the shape of the distribution for the cloudy states, depend characteristically on the average degree of cloudiness, represented by the daily clear-sky index. For dispersed sites within the studied network, separated by distances up to 1 km, and for daily clear-sky indices above approximately 0.5, the model performs better in reproducing the aggregate clear-sky index than non-spatial data. The proposed model could assist distribution system operators (DSOs) in grid planning and operation, as shown in a case study.