  We prove that if $S$ is a smooth reflexive surface in $\mathbb{P}^3$ defined
over a finite field $\mathbb{F}_q$, then there exists an $\mathbb{F}_q$-line
meeting $S$ transversely provided that $q\geq c\operatorname{deg}(S)$, where
$c=\frac{3+\sqrt{17}}{4}\approx 1.7808$. Without the reflexivity hypothesis, we
prove the existence of a transverse $\mathbb{F}_q$-line for $q\geq
\operatorname{deg}(S)^2$.
