We report on a new measurement of the beam transverse single spin asymmetry in electron-proton elastic scattering, A_{⊥}^{ep}, at five beam energies from 315.1 to 1508.4 MeV and at a scattering angle of 30°<θ<40°. The covered Q^{2} values are 0.032, 0.057, 0.082, 0.218, 0.613 (GeV/c)^{2}. The measurement clearly indicates significant inelastic contributions to the two-photon-exchange (TPE) amplitude in the low-Q^{2} kinematic region. No theoretical calculation is able to reproduce our result. Comparison with a calculation based on unitarity, which only takes into account elastic and πN inelastic intermediate states, suggests that there are other inelastic intermediate states such as ππN, KΛ, and ηN. Covering a wide energy range, our new high-precision data provide a benchmark to study those intermediate states.