Nonlinear Compton scattering is calculated for the collision of an electron with a plane-wave pulse. A midinfrared (IR) peak arises in the photon spectrum due to long-range interference associated with the pulse envelope. The case of a flattop pulse is studied as a toy model for pulse envelope effects and reduced to two final-state momentum integrations; the case of a sine-squared pulse is studied numerically. A perturbative expansion in charge-field coupling reveals that already at intermediate intensities, many orders are required to correctly capture the structure of the mid-IR peak. By regularizing the classical result, it is shown that the mid-IR peak is due to plane-wave ponderomotive effects from the pulse envelope. Finally, it is shown that the mid-IR peak can be isolated using energy, angle and polarization filters.