Force appropriation testing has long been used for ground vibration testing of aircraft, where it is critical to estimate the modal parameters and especially damping accurately. Recently, extensions were presented that allow systematic identification of the nonlinear normal modes (NNMs) of conservative and non-conservative nonlinear structures. While this method provides accurate results with high confidence, it is unfortunately quite slow and so the structure may be subjected to significant damage over the course of a test. This work proposes a new approach in which the test is performed more quickly by simply acquiring measurements near the nonlinear resonance, but without the time consuming tuning required to reach the resonance precisely. Then, the recently proposed single nonlinear resonant mode method is used to interpolate between test points in order to estimate the NNM from each set of forced responses. The method is first evaluated numerically using a reduced model of a curved clamped-clamped beam that exhibits both softening and hardening response due to geometric nonlinearity. Then the method is employed experimentally to measure the first two NNMs of a curved beam that was manufactured from plastic using a 3D printer and the results are compared to the traditional tuning approach.