  In this article we show that part of the argument used in
\cite{de_ambroggio_roberts1} to derive the correct upper bound for the
probability of observing an unusually large maximal component in the
near-critical Erd\H{o}s-R\'enyi graph can be used to analyse other models of
random graphs when considered at criticality. Specifically, we apply our method
to a model of random intersection graph, a random graph obtained through
$p$-bond percolation on a general $d$-regular graph, and a model of
inhomogeneous random graph.
