In exterior acoustic simulations with the finite element method, accurately modeling an infinite domain using a finite computational space is challenging due to reflections at the truncated boundaries. This study introduces an m-th order operator for implementing absorbing boundary conditions that releases the geometrical constraints and minimizes reflections. Furthermore, the resulting finite element problem is naturally well-suited for a range of reduced-order models, such as the moment-matching, projection based well-conditioned asymptotic waveform evaluation (WCAWE), allowing efficient frequency sweep studies in large models. Our combined approach significantly enhances simulation efficiency, allowing for extensive frequency analysis with minimal domain size without compromising accuracy. However, the combination of higher order formulations of the considered absorbing boundary condition with the WCAWE approach also exhibits limitations in accuracy for a given size of reduced basis. This is aspect is the object of ongoing investigations.