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 $ζ\in(0,1)$ 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 \[ \mathbb{P}(k< |\mathcal{C}(0)|<\infty)=\exp\big(-Θ(k^ζ)\big), \qquad \text{as }k\to\infty. \] The value of $ζ$ undergoes several phase transitions with respect to three main model parameters: the Euclidean dimension $d$, the power-law tail exponent $τ$ of the degree distribution and a long-range parameter $α$ 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: $ζ$ in this regime depends both on $τ,α$. We also prove that the second-largest component in a box of volume $n$ is of size $Θ((\log n)^{1/ζ})$ 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.