  For graphs $G$ and $H$, we write $G \overset{\mathrm{rb}}{\longrightarrow} H
$ if any proper edge-coloring of $G$ contains a rainbow copy of $H$, i.e., a
copy where no color appears more than once. Kohayakawa, Konstadinidis and the
last author proved that the threshold for $G(n,p)
\overset{\mathrm{rb}}{\longrightarrow}H$ is at most $n^{-1/m_2(H)}$. Previous
results have matched the lower bound for this anti-Ramsey threshold for cycles
and complete graphs with at least 5 vertices. Kohayakawa, Konstadinidis and the
last author also presented an infinite family of graphs $H$ for which the
anti-Ramsey threshold is asymptotically smaller than $n^{-1/m_2(H)}$. In this
paper, we devise a framework that provides a richer and more complex family of
such graphs that includes all the previously known examples.
