In the face of vaccine dose shortages and logistical challenges, various deployment strategies are being proposed to increase population immunity levels to SARS-CoV-2. Two critical issues arise: how will the timing of delivery of the second dose affect both infection dynamics and prospects for the evolution of viral immune escape via a build-up of partially immune individuals. Both hinge on the robustness of the immune response elicited by a single dose, compared to natural and two-dose immunity. Building on an existing immuno-epidemiological model, we find that in the short-term, focusing on one dose generally decreases infections, but longer-term outcomes depend on this relative immune robustness. We then explore three scenarios of selection and find that a one-dose policy may increase the potential for antigenic evolution under certain conditions of partial population immunity. We highlight the critical need to test viral loads and quantify immune responses after one vaccine dose, and to ramp up vaccination efforts throughout the world.