Extremal values for the sum $\sum_{r=1}^\tau e(a 2^r/q)$

Let $q$ be an odd integer, let $\tau$ be the order of $2$ modulo $q$, and let $a$ be coprime with $q$. Finally, let $s(a/q) := \sum_{r=1}^\tau e(a 2^r/q)$. We prove that $|s(a/q)|$ can be as large as $\tau-c'$ for a suitable constant $c'$ for infinitely many $q$, but that $\max_{(a,q)=1}|s(a/q)|$ is always bounded from above by $\tau-c$ for a suitable positive constant $c$ whose value is considerably larger that any previously known. An upper bound for $\min_{(a,q)=1}|s(a/q)|$ is also discussed.