  We evaluate the pion-nucleon intermediate-state contribution to the
two-photon exchange (TPE) correction in the elastic electron-nucleon scattering
within a dispersive framework. We calculate the contribution from all $\pi N$
partial waves using the MAID parametrization. We provide the corresponding TPE
correction to the unpolarized $e p$ scattering cross section in the region of
low momentum transfer $ Q^2 \lesssim 0.064~\mathrm{GeV}^2$, where no analytical
continuation into the unphysical region of the TPE scattering amplitudes is
required. We compare our result in the forward angular region with an
alternative TPE calculation, in terms of structure functions, and find a good
agreement, indicating a small contribution at low $Q^2$ due to discontinuities
beyond $ \pi N$. We also compare our results with empirical fits.
