  In a recent paper we have considered the long time asymptotics of the
periodic Toda lattice under a short range perturbation and we have proved that
the perturbed lattice asymptotically approaches a modulated lattice. In the
present paper we capture the higher order asymptotics, at least away from some
resonance regions. In particular we prove that the decay rate is $O(t^{-1/2})$.
  Our proof relies on the asymptotic analysis of the associated Riemann-Hilbert
factorization problem, which is here set on a hyperelliptic curve. As in
previous studies of the free Toda lattice, the higher order asymptotics arise
from "local" Riemann-Hilbert factorization problems on small crosses centered
on the stationary phase points. We discover that the analysis of such a local
problem can be done in a chart around each stationary phase point and reduces
to a Riemann--Hilbert factorization problem on the complex plane. This result
can then be pulled back to the hyperelliptic curve.
