Character covering number of $\mathrm{PSL}_2 (q)$

For a group $G$ and a character $\chi$ of $G$, let $c(\chi)$ denote the set of all irreducible characters of $G$, occurring in $\chi$. The character covering number of $G$ is defined as the least $n$ such that $ c(\chi^n)=\operatorname{Irr}(G)$, for all faithful irreducible $\chi$. In this article, we compute the character covering number of $\mathrm{PSL}_2(q)$ for all $q\geq8$.