The $G^0$-action on a coset is the same as a so-called twisted action which is pretty well understood. See, e.g., Mohrdieck, S.: Conjugacy classes of non-connected semisimple algenraic groups, Transformation Groups, 8, (2003)
377-395.

More precisely, let $C=G^0a$ be a connected component of $G$. Then conjugation by $a$ induces an automorphism $\tau$ on $G^0$. Identifying $C$ with $G^0$ via $g\mapsto ga$ converts conjugation on $C$ to twisted conjugation on $G^0$
$$
u(ga)u^{-1}=(ug\tau(u)^{-1})a
$$
It is possible to choose $a$ in such a way that $\tau$ preserves a Borel $B$, a maximal torus $T$ and a pinning of $G^0$, i.e., $\tau$ is induced by an automorphism of the Dynkin diagram of $G^0$. Let $T^0:=(T^\tau)^0$. Then it is not difficult to see that the map $G^0\times T_0\to G^0:(g,t)\mapsto gt\tau(g)^{-1}$ is dominant. Thus the conjugacy classes of $T_0a$ contain an open subset of $C$ and they are all semisimple.