  We consider discrete Schr{\"o}dinger operators on ${\mathbb{Z}}^d$ for which
the perturbation consists of the sum of a long-range type potential and a
Wigner-von Neumann type potential. Still working in a framework of weighted
Mourre theory, we improve the limiting absorption principle (LAP) that was
obtained in [Ma1]. To our knowledge, this is a new result even in the
one-dimensional case. The improvement consists in a weakening of the
assumptions on the long-range potential and better LAP weights. The improvement
relies only on the fact that the generator of dilations (which serves as
conjugate operator) is bounded from above by the position operator. To exploit
this, Loewner's theorem on operator monotone functions is invoked.
