We show how a one-fluid polytropic solar wind model exhibits properties similar to an isothermal wind when localized momentum addition and/or rapid area divergence produce multiple critical points in the flow. In particular, we show that when the sonic transition in the flow occurs closer to the coronal base, multiple steady solutions can exist. These multiple steady solutions consist of a continuous solution passing through the innermost critical point and other steady solutions involving a steady shock transition. By following the temporal evolution of the solar wind from a steady state with one critical point to a steady state with three critical points, we show that a standing shock solution is more likely to develop than a continuous solution when momentum deposition occurs close to the coronal base and the equation of motion admits multiple steady solutions. This result is particularly relevant to the solar wind when momentum deposition occurs as a result of a rapidly diverging coronal hole geometry.