  We consider a family of linear singularly perturbed Cauchy problems which
combines partial differential operators and linear fractional transforms. We
construct a collection of holomorphic solutions on a full covering by sectors
of a neighborhood of the origin in $\mathbb{C}$ with respect to the
perturbation parameter $\epsilon$. This set is built up through classical and
special Laplace transforms along piecewise linear paths of functions which
possess exponential or super exponential growth/decay on horizontal strips. A
fine structure which entails two levels of Gevrey asymptotics of order 1 and
so-called order $1^{+}$ is witnessed. Furthermore, unicity properties regarding
the $1^{+}$ asymptotic layer are observed and follow from results on
summability w.r.t a particular strongly regular sequence recently obtained in a
previous study.
