Because of the seraicollisional nature of the solar wind, the collisionless or exospheric approach and the hydrodynamic one are both inaccurate. However, the advantage of simplicity makes them useful for enlightening us on some basic mechanisms of solar wind acceleration. Previous exospheric models have been able to reproduce winds that were already nearly supersonic at the exobase, the altitude above which there are no collisions. In order to allow transonic solutions, a lower exobase has to be considered, in which case the protons are experiencing a nonmonotonic potential energy profile. This is done in the present work. In this model, the electron velocity distribution in the corona is assumed to be nonthermal. Parametric results are presented and show that the high acceleration obtained does not depend on the details of the nonthermal distributions. This acceleration seems, therefore, to be a robust result produced by the presence of a sufficient number of suprathermal electrons. A method for improving the exospheric description is also given, which consists of mapping particle orbits in terms of their invariants of motion.