[Background] Above the nucleon resonance region, the $N(e,epi^pm)N$ data cannot be explained by conventional hadronic models. For example, the observed magnitude of the transverse cross section is significantly underestimated in a framework with Reggeized background amplitudes. [Purpose] Develop a phenomenological framework for the $N(e,epi^pm)N$ reaction at high invariant mass $W$ and deep photon virtuality $Q^2$. [Method] Building on the work of Kaskulov and Mosel, a gauged pion-exchange current is introduced with a running cutoff energy for the proton electromagnetic transition form factor. A new transition form factor is proposed. It respects the correct on-shell limit, has a simple physical interpretation and reduces the number of free parameters by one. [Results] A study of the $W$ dependence of the $N(e,epi^pm)N$ lends support for the newly proposed transition form factor. In addition, an improved description of the separated and unseparated cross sections at $-t lesssim 0.5 ;text{GeV}^2$ is obtained. The predictions overshoot the measured unseparated cross sections for $-t > 0.5 ;text{GeV}^2$. Introducing a strong hadronic form factor in the Reggeized background amplitudes brings the calculations considerably closer to the high $-t$ data. [Conclusions] Hadronic models corrected for resonance/parton duality describe the separated pion electroproduction cross sections above the resonance region reasonably well at low $-t$. In order to validate the applicability of these models at high $-t$, separated cross sections are needed. These are expected to provide a more profound insight into the relevant reaction mechanisms.
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