Let $G=(V,E)$ be a finite, simple, undirected graph. The Hadwiger number $\eta(G)$ of $G$ is defined to be the largest positive integer $n\in\mathbb{N}$ such that the complete graph $K_n$ is a minor of $G$.
We say that a graph is vertex-critical if removing any vertex reduces the chromatic number.
Question. What is an example of a connected, non-complete vertex-critical graph $G=(V,E)$ with the following property?
Adding any edge increases the Hadwiger number -- or more formally: for $v\neq w \in V$ with $\{v,w\}\notin E$ we get $$\eta\big((V,E\cup\big\{\{v,w\}\big\})\big) > \eta\big((V,E)\big).$$
Motivation. Any minimal counterexample to Hadwiger's conjecture has the properties given in the question.
