It is known that given any smooth ($C^\infty$) simple closed curve $\gamma$ in the 2-sphere, there is a smooth diffeomorphism of $\mathbb{S}^2$ taking $\gamma$ to the standard equator $x^2+y^2=1$.
Question: if $\gamma$ is instead piecewise smooth, it is there a homeomorphism of $\mathbb{S}^2$ taking $\gamma$ to the standard equator, whose restriction to the complement in $\mathbb{S}^2$ of the set of non-smooth points of $\gamma$ is a smooth diffeomorphism?
References would be much appreciated.