Complete minors and stability numbers

Hadwiger's conjecture implies that $n\le\alpha h$ for all graphs of order $n$, stability number $\alpha$, and Hadwiger number $h$. Combining ideas of Kawarabayashi et al. and Wood, we prove that $n\le (\alpha-1)(2h-5)+5$ for such graphs if $\alpha\ge 3$ and $h\ge 5$.