  We consider a large class of spatially-embedded random graphs that includes
among others long-range percolation, continuum scale-free percolation and the
age-dependent random connection model. We assume that the model is
supercritical: there is an infinite component. We identify the stretch-exponent
$\zeta\in(0,1)$ of the decay of the cluster-size distribution. That is, with
$|\mathcal{C}(0)|$ denoting the number of vertices in the component of the
vertex at $0\in \mathbb{R}^d$, we prove \[
  \mathbb{P}(k< |\mathcal{C}(0)|<\infty)=\exp\big(-\Theta(k^{\zeta})\big),
\qquad \text{as }k\to\infty.
  \] The value of $\zeta$ undergoes several phase transitions with respect to
three main model parameters: the Euclidean dimension $d$, the power-law tail
exponent $\tau$ of the degree distribution and a long-range parameter $\alpha$
governing the presence of long edges in Euclidean space. In this paper we
present the proof for the region in the phase diagram where the model is a
generalization of continuum scale-free percolation and/or hyperbolic random
graphs: $\zeta$ in this regime depends both on $\tau,\alpha$. We also prove
that the second-largest component in a box of volume $n$ is of size
$\Theta((\log n)^{1/\zeta})$ with high probability. We develop a deterministic
algorithm, the cover expansion, as new methodology. This algorithm enables us
to prevent too large components that may be de-localized or locally dense in
space.
