  We study the giant component problem slightly above the critical regime for
percolation on Poissonian random graphs in the scale-free regime, where the
vertex weights and degrees have a diverging second moment. Critical percolation
on scale-free random graphs have been observed to have incredibly subtle
features that are markedly different compared to those in random graphs with
converging second moment. In particular, the critical window for percolation
depends sensitively on whether we consider single- or multi-edge versions of
the Poissonian random graph.
  In this paper, and together with our companion paper with Bhamidi, we build a
bridge between these two cases. Our results characterize the part of the barely
supercritical regime where the size of the giant components are approximately
same for the single- and multi-edge settings. The methods for establishing
concentration of giant for the single- and multi-edge versions are quite
different. While the analysis in the multi-edge case is based on scaling limits
of exploration processes, the single-edge setting requires identification of a
core structure inside certain high-degree vertices that forms the giant
component.
