On additive bases of finite groups

Let $G$ be a multiplicatively written finite group. The critical number $\mathsf{cr}(G)$ of $G$ is the smallest integer $t$ such that for every subset $S$ of $G\setminus \{1\}$ with $|S|\geq t$ the following holds: every element of $G$ can be written as a non-empty product of distinct elements from $S$. We prove that $\mathsf{cr}(G)\leq |G|/p+p-2$ for all finite non-abelian groups $G$ with $|G|\neq 6$, where $p$ is the smallest prime divisor of $|G|$. Moreover, equality holds if and only if $G$ has a subgroup of index $p$.