The hybrid finite element-boundary integral (FE-BI) method has been quite popular due to its adaptability in modeling arbitrarily shaped objects involving complex materials. However, being a frequency domain analysis, the FE-BI method may not be appealing for broadband frequency-dependent data generation. Also, it is unavoidable to resolve the hybrid system for each excitation (incidence angle) when an iterative solver is employed. Subsequently, monostatic radar cross section (RCS) calculations are computationally intensive, and therefore asymptotic waveform evaluation (AWE) as an extrapolation method becomes attractive for CPU time reduction. In this paper, we present frequency and angular extrapolation implementations in FE-BI systems. Scattering applications of the hybrid technique in conjunction with AWE will be presented to demonstrate the performance of these implementations in terms of accuracy and computational efficiency.