  In this paper we provide a procedure to obtain a non-trivial HHS structure on
a hyperbolic space. In particular, we prove that given a finite collection
$\mathcal{F}$ of quasi-convex subgroups of a hyperbolic group $G$, there is an
HHG structure on $G$ that is compatible with $\mathcal{F}$. We will use this to
provide explicit descriptions of the Gromov Boundary of hyperbolic HHS and HHG,
and we recover results from Hamenst\"adt, Manning, Trang for the case when $G$
is hyperbolic relative to $\mathcal{F}$. Further applications in the
construction of new HHG will be presented in a subsequent paper.
