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 $T_0:=(T^\tau)^0$ be the connected component of the $\tau$-fixed points in $T$. 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.

**Edit:** Urs Hartl pointed out to me that the proof of Prop. 3.8 in *loc.cit* may contain a gap (it is unclear that $t$ exists such that $t^{\mathrm ord\,\tau}$ is regular semisimple). Therefore, I am adding a direct argument for the claim that $G^0\times T_0\to G^0$ is dominant. This is done in two stages: Let $C:=(G^\tau)^0\subseteq G^0$. First one shows that
$$
G^0\times C\to G^0:(g,c)\mapsto gc\tau(g)^{-1}
$$
is dominant. For this it suffices to show that the map on tangent spaces in $(1,1)$ is surjective. Because of $\mathrm{Lie}\,C=\ker(1-\tau)$ that follows from
$$
(1-\tau)\mathfrak g\oplus \ker(1-\tau)=\mathfrak g
$$
(observe that $\tau$ is of finite order). So all elements in an open subset of $G^0$ are twisted conjugate to an element of $C$. For the second step observe that twisted conjugation on $C$ is ordinary conjugation and that $T_0$ is a maximal torus of $C$. So all elements in an open subset of $C$ are (twisted) conjugate to an element of $T_0$. q.e.d