A new class of shock transitions are shown to arise in the transonic solutions of the steady isothermal solar wind equations when, for example, momentum deposition gives rise to multiple critical points in the flow. These shock transitions between critical solutions occur for a certain range of the parameters characterizing the momentum deposition function. In the presence of such shock transitions, the isothermal wind equations admit multiple transonic solutions, namely a continuous solution passing through an inner critical point and solutions involving a shock transition between critical solutions. These multiple transonic solutions have the same flow speed at the base but different supersonic flow speeds at infinity. An additional interesting feature of the isothermal equations is the equivalence of nonradial flow tube divergence and momentum addition in giving rise to multiple critical points and hence to multiple transonic solutions with shock transitions. The physical relevance of these properties for astrophysical systems such as the inner solar wind, flows in extragalactic jets and accretion discs are discussed.