  Migdal-Eliashberg theory is one of the state-of-the-art methods for
describing conventional superconductors from first principles. However, widely
used implementations assume a constant density of states around the Fermi
level, which hinders a proper description of materials with distinct features
in its vicinity. Here, we present an implementation of the Migdal-Eliashberg
theory within the EPW code that considers the full electronic structure and
accommodates scattering processes beyond the Fermi surface. To significantly
reduce computational costs, we introduce a non-uniform sampling scheme along
the imaginary axis. We demonstrate the power of our implementation by applying
it to the sodalite-like clathrates YH$_6$ and CaH$_6$, and to the
covalently-bonded H$_3$S and D$_3$S. Furthermore, we investigate the effect of
maximizing the density of states at the Fermi level in doped H$_3$S and
BaSiH$_8$ within the full-bandwidth treatment compared to the
constant-density-of-states approximation. Our findings highlight the importance
of this advanced treatment in such complex materials.
