Covering finite fields by translates of multiplicative subgroups

Let $G_d$ be the subgroup of index $d>1$ in $\mathbb{F}_q^\times$, and put $G_d^0=G_d\cup\{0\}$. For a covering $A+G_d^0=\mathbb{F}_q$ with $\#A=s$, we prove $q\le d^{8s/d}$ by shifting the roots of a Bonferroni polynomial. Although a uniform bound in $s$ alone must allow exponential growth, our est