  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.
