  In this work, the two-photon-exchange (TPE) effect in $ep\rightarrow en\pi^+$
at small $-t$ is discussed. In the previous work, the TPE contribution with one
$\pi$ intermediate state is estimated numerically within a hadronic model under
the pion-dominance approximation. Here we extend the discussion to include one
$\rho$ intermediate state. The TPE contribution can be described by one scalar
function in the limit $m_e\rightarrow 0$, the dispersion relation (DR)
satisfied by this scalar function is analysed. The analytic expressions for the
imaginary parts of the TPE contributions from one $\pi$ or one $\rho$
intermediate state are given within the hadronic model. Combining these
analytic expressions and the DR, the corresponding real parts of the TPE
contributions can be estimated easily at any available region. This can help
the further experimental analysis to include the TPE contributions in a
convenient way. The numeric results show that the TPE correction with one
$\rho$ intermediate state is much smaller than that with one $\pi$ intermediate
state in the current energy region. These results suggest that the TPE
contribution with an elastic state is the main TPE contribution in
$ep\rightarrow en\pi^+$ at small $-t$.
