  Blazars exhibit flares across the entire electromagnetic spectrum. Many
$\gamma$-ray flares are highly correlated with flares detected at longer
wavelengths; however, a small subset appears to occur in isolation, with little
or no correlated variability at longer wavelengths. These "orphan" $\gamma$-ray
flares challenge current models of blazar variability, most of which are unable
to reproduce this type of behavior. Macdonald et al. have developed the Ring of
Fire model to explain the origin of orphan $\gamma$-ray flares from within
blazar jets. In this model, electrons contained within a blob of plasma moving
relativistically along the spine of the jet inverse-Compton scatter synchrotron
photons emanating off of a ring of shocked sheath plasma that enshrouds the jet
spine. As the blob propagates through the ring, the scattering of the ring
photons by the blob electrons creates an orphan $\gamma$-ray flare. This model
was successfully applied to modeling a prominent orphan $\gamma$-ray flare
observed in the blazar PKS 1510$-$089. To further support the plausibility of
this model, Macdonald et al. presented a stacked radio map of PKS 1510$-$089
containing the polarimetric signature of a sheath of plasma surrounding the
spine of the jet. In this paper, we extend our modeling and stacking techniques
to a larger sample of blazars: 3C 273, 4C 71$.$01, 3C 279, 1055$+$018, CTA 102,
and 3C 345, the majority of which have exhibited orphan $\gamma$-ray flares. We
find that the model can successfully reproduce these flares, while our stacked
maps reveal the existence of jet sheaths within these blazars.
