A widely applied dose-response model is the Hill model with four parameters. In the Hill model, the measurement error may be homoscedastic or heteroscedastic. If the experimental points are fixed with the goal of identifying the right error-variance structure, then by applying the KL-criterion, we have that only two different doses are necessary. Hence, this optimum design does not enable any estimation of the parameters. From here the necessity to add some other experimental points. This work focuses on augmenting the KL-optimal design by the inclusion of two additional doses with the goal of providing an estimation of the model parameters, while guaranteeing a minimum KL-efficiency to optimally discriminate between the two variance structures.