In the present work, the kinetic exospheric model of the solar wind is improved by considering regularized Kappa distributions that have no diverging moments through consideration of a cut-off at relativistic velocities. The model becomes valid even for kappa indices lower than 2, which is important since low values of kappa are observed in the fast solar wind. The exospheric model shows that the electric potential accelerates the wind to supersonic velocities. The presence of suprathermal Strahl electrons at the exobase can further increase the velocity to higher values, leading to profiles comparable to the observations in the fast and slow wind at all radial distances. The kappa index is not the only parameter that influences the acceleration of the wind: the difference in the altitude of the exobase also makes a significant difference between the fast and slow wind. The exobase is located at lower altitudes in the coronal holes where the density is smaller than in the other regions of the corona, allowing the wind originating from the holes to be accelerated to higher velocities. The new observations of Parker Solar Probe are used to constrain the model. The observations at low radial distances show suprathermal electrons already well present in the Strahl in the antisunward direction and a deficit in the sunward direction, confirming the exospheric feature of almost no incoming particles. For proton distributions, we observe that the proton tail parallel to the magnetic field is already present at 17.2 Rs.