Let $X$ be a smooth projective surface (other than the projective plane) and $E$ be a rank $r \geq 2$ globally generated vector bundle over $X$ such that $\text{det}(E) \cong L$. If $H^0(X, E) \neq 0$ and $H^0(X, E\otimes L^*)=0$, then can we determine some specific properties of $E$?

For example can we deduce that $E$ is decomposable?

Any suggestion (including some specific cases e.g. $r=2$ over some specific surface) are welcome.