By means of a nonlinear separation of the variables in the governing full set of the magnetohydrodynamic (MHD) equations for axisymmetric plasmas we analyse an exact model for magnetized and rotating outflows which are hotter and overpressured at their axis. These outflows start subsonically and subAlfvénically from the central gravitating source and its surrounding accretion disk. Subsequently, they accelerate thermally and magnetocentrifugally and thus cross the appropriate MHD critical points, reaching high values of the Alfvén Mach number. Three types of solutions are found : (a) collimated jet-type outflows from efficient magnetic rotators with the flow confined by the magnetic hoop stress; (b) radially expanding wind-type outflows analogous to the solar wind, from inefficient magnetic rotators or strongly overpressured sources; (c) terminated solutions with increasing amplitude of oscillations in the width of the beam. In contrast to previously studied underpressured outflows, the transition from collimated jets to uncollimated winds is not continuous in the appropriate parametric space with a gap where no stationary solution is found. Superfast at infinity solutions are filtered by three critical surfaces corresponding to the three known limiting characteristics or separatrices of MHD wind theory. Collimated and terminated solutions cross the slow, Alfvén and fast magneto-acoustic critical points. Radially expanding solutions cross the slow and Alfvén critical points while the last boundary condition is imposed by requiring that the pressure vanishes at infinity.