Abstract Statistical classification of the Helios solar wind observations into several populations sorted by bulk speed has revealed an outward acceleration of the wind. The faster the wind, the smaller this acceleration in the 0.3–1 au radial range. In this paper, we show that recent measurements from the Parker Solar Probe (PSP) are compatible with an extension closer to the Sun of the latter Helios classification. For instance, the well-established bulk speed/proton temperature ( u , T p ) correlation and bulk speed/electron temperature ( u , T e ) anticorrelation, together with the acceleration of the slowest winds, are verified in PSP data. We also model the combined PSP and Helios data using empirical Parker-like models for which the solar wind undergoes an “isopoly” expansion: isothermal in the corona, then polytropic at distances larger than the sonic point radius. The polytropic indices are derived from the observed temperature and density gradients. Our modeling reveals that the electron thermal pressure has a major contribution in the acceleration process of slow and intermediate winds (in the range of 300–500 km s −1 at 1 au) over a broad range of distances and that the global (electron and proton) thermal energy alone is able to explain the acceleration profiles. Moreover, we show that the very slow solar wind requires, in addition to the observed pressure gradients, another source of acceleration.