We present a new approach to the  eigensystem multiscale analysis (EMSA) for the Anderson model that relies on the Wegner estimate.  The EMSA  treats all energies  of the finite volume operator in an energy interval at the same time,  simultaneously establishing localization of  all eigenfunctions with eigenvalues in the energy interval  with high probability.  It implies all the usual manifestations of localization (pure point spectrum with exponentially decaying eigenfunctions, dynamical localization)   for the Anderson model. The new method removes the restrictive level spacing hypothesis used in the previous versions of the EMSA, allowing  for single site probability distributions that are  H\"older continuous of order  $\alpha \in (0,1]$. (Joint work with Alex Elgart.)