  We investigate the onset of the discontinuous percolation transition in
small-world hyperbolic networks by studying the systems-size scaling of the
typical largest cluster approaching the transition, $p\nearrow p_{c}$. To this
end, we determine the average size of the largest cluster $\left\langle s_{{\rm
max}}\right\rangle \sim N^{\Psi\left(p\right)}$ in the thermodynamic limit
using real-space renormalization of cluster generating functions for bond and
site percolation in several models of hyperbolic networks that provide exact
results. We determine that all our models conform to the recently predicted
behavior regarding the growth of the largest cluster, which found diverging,
albeit sub-extensive, clusters spanning the system with finite probability well
below $p_{c}$ and at most quadratic corrections to unity in
$\Psi\left(p\right)$ for $p\nearrow p_{c}$. Our study suggest a large
universality in the cluster formation on small-world hyperbolic networks and
the potential for an alternative mechanism in the cluster formation dynamics at
the onset of discontinuous percolation transitions.
