  We calculate the two-photon exchange (TPE) corrections in the region where
the kinematical variables describing the elastic $ep$ scattering are moderately
large momentum scales relative to the soft hadronic scale. For such kinematics
we use the QCD factorization approach formulated in the framework of the
soft-collinear effective theory (SCET). Such technique allows us to develop a
description for the soft-spectator scattering contribution which is found to be
important in the region of moderately large scales.
  Together with the hard-spectator contribution we present the complete
factorization formulas for the TPE amplitudes at the leading power and leading
logarithmic accuracy. The momentum region where both photons are hard is
described by only one new nonperturbative SCET form factor. It turns out that
the same form factor also arises for wide-angle Compton scattering which is
also described in the framework of the SCET approach. This allows us to
estimate the soft-spectator contribution associated with the hard photons in a
model independent way.
  The main unknown in our description of the TPE contribution is related with
the configuration where one photon is soft. The nonperturbative dynamics in
this case is described by two unknown SCET amplitudes. We use a simple model in
order to estimate their contribution.
  The formalism is then applied to a phenomenological analysis of existing data
for the reduced cross section as well as for the transverse and longitudinal
polarization observables.
