Surface-plasmon resonance (SPR) is a sensing technique widely used for its label-free feature. However, its sensitivity is contingent on the divergence angle of the excitation beam. The problem becomes pronounced for compact systems when a low-cost LED is used as the light source. When the Kretschmann configuration with a periodically modulated surface is used, a bandgap appears in the surface plasmon dispersion relation. We recognize that the high density of modes on the edge of the surface-plasmon bandgap permits the coupling of a wider range of incidence angles of excitation photons to surface-plasmon polaritons than what is possible in the traditional Kretschmann configuration. Here, the numerical simulation illustrates that the sensitivity, detection limit, and reflectivity minimum of an amplitude-based SPR bandgap-assisted surface-plasmon sensor are almost independent of the divergence angle. Two different bandgap structures are compared with the Kretschmann configuration using the rigorous coupled-wave analysis technique. The results indicate that the bandgap-assisted sensing outperforms traditional SPR sensing when the angular standard deviation of the excitation beam is above 1 degree.