  We study the large $N$ expansion of winding Wilson loops in the off-critical
regime of the Gross-Witten-Wadia (GWW) unitary matrix model. These have been
recently considered in arXiv:1705.06542 and computed by numerical methods. We
present various analytical algorithms for the precise computation of both the
perturbative and instanton corrections to the Wilson loops. In the gapped phase
of the GWW model we present the genus five expansion of the one-cut resolvent
that captures all winding loops. Then, as a complementary tool, we apply the
Periwal-Shevitz orthogonal polynomial recursion to the GWW model coupled to
suitable sources and show how it generates all higher genus corrections to any
specific loop with given winding. The method is extended to the treatment of
instanton effects including higher order $1/N$ corrections. Several explicit
examples are fully worked out and a general formula for the next-to-leading
correction at general winding is provided. For the simplest cases, our
calculation checks exact results from the Schwinger-Dyson equations, but the
presented tools have a wider range of applicability.
