  We show that if $G_1$ and $G_2$ are non-solvable groups, then no $C^{1,\tau}$
action of $(G_1\times G_2)*\mathbb{Z}$ on $S^1$ is faithful for $\tau>0$. As a
corollary, if $S$ is an orientable surface of complexity at least three then
the critical regularity of an arbitrary finite index subgroup of the mapping
class group $\mathrm{Mod}(S)$ with respect to the circle is at most one, thus
strengthening a result of the first two authors with Baik.
