Let $E$ be a locally free sheaf on a complete nonsingular curve $C$ over $\mathbb C$.
Suppose that, for all points $P$ in $C$, the locally free sheaf $$ E\otimes \mathcal{O}_C(P)$$ is an ample locally free sheaf on $C$. Also, suppose that $E$ is not isomorphic to $\mathcal{O}_C^{r}$, where $r$ is the rank of $E$.
Does it follow that $E$ is ample?
It's true when $r=1$.
I think it should be true, but I don't see how to do it. Intuitively, I keep thinking about the $r=1$ case. In that case you could just compute the degree, but you could also use that for all rational sections of $E$ there exists a point $P$ not in the support of the divisor of that section. This helps you conclude that the divisor of your section is effective, and thus ample. Does this line of thought help when $r>1$?
Final remark. By induction, we may and do assume $E$ is indecomposable. In particular, it's true for all $r\geq 1$ when $C =\mathbf P^1$.
Edit: Asal shows that it is not true under the above conditions when $g=1$. Let me show that it is also false when $g\geq 2$.
Let $$ 0 \to \omega_C\to E\to \mathcal O_C\to 0$$ be a non-trivial extension $\mathcal O_C$ by $\omega_C$. This exists because $H^1(\omega_C) \cong \mathbb C \neq 0$. Then $E$ is not ample, because $E$ surjects onto $\mathcal O_C$. But clearly, $E\otimes L$ is an extension of $L$ by $L\otimes \omega_C$. If $L$ is ample, then $L\otimes \omega_C$ is ample. In particular, $E\otimes L$ is an extension of an ample line bundle by an ample line bundle. Thus, $E\otimes L$ is ample. (Now apply this to $L=\mathcal {O}_C(P)$...)