  We consider a $2d$ magnetic Schr\"odinger operator perturbed by a weak
magnetic field which slowly varies around a positive mean. In a previous paper
we proved the appearance of a `Landau type' structure of spectral islands at
the bottom of the spectrum, under the hypothesis that the lowest Bloch
eigenvalue of the unperturbed operator remained simple on the whole Brillouin
zone, even though its range may overlap with the range of the second
eigenvalue. We also assumed that the first Bloch spectral projection was smooth
and had a zero Chern number.
  In this paper we extend our previous results to the only two remaining
possibilities: either the first Bloch eigenvalue remains isolated while its
corresponding spectral projection has a non-zero Chern number, or the first two
Bloch eigenvalues cross each other.
