Vapor pressure of pure substances is a crucial piece of information for many industrial applications. A recently developed PR+COSMOSAC equation of state (EOS) with its molecular interaction parameters determined from quantum mechanical calculations has been shown to provide reasonable prediction accuracy for vapor pressure of almost any chemical species without the issue of missing parameters. In this work, we introduce two modifications to improve the accruacty of its prediction of the vapor pressure. The overall deviation in pressure for 1124 pure liquids from the modified version is 138%, which is about 1/4 of that of the original model (553%). In particular, the accuracy for the vapor pressure near the triple point shows major improvement, with the average error reduced from 1062% to 233%. The sublimation pressure can also be estimated, provided that the melting temperature and enthalpy of fusion are available. The average deviation in sublimation pressure from the modified PR+COSMOSAC EOS for 1140 substances is 412%, which is only 1/3 that of the original model (1249%). This model is capable of providing both the vapor pressure and sublimation pressure over a wide range of conditions (from the critical point to below the triple point). It is particularly useful when experimental data are not available.