We report on a new measurement of the beam transverse single spin asymmetry in electron-proton elastic scattering, $A^{ep}_{perp}$, at five beam energies from 315.1 MeV to 1508.4 MeV and at a scattering angle of $30^{circ} < theta < 40^{circ}$. 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 $mathrm{pi N}$ inelastic intermediate states, suggests that there are other inelastic intermediate states such as $mathrm{pi pi N}$, $mathrm{K Lambda}$ and $mathrm{eta N}$. Covering a wide energy range, our new high-precision data provide a benchmark to study those intermediate states.
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