You should probably state what you want a little more precisely. As it is currently stated, it allows for the possibility that $\pi$ is not projective in which case there is no chance. The statement also allows a small resolution in which case $E$ is empty and $a\pi^*L$ is not ample for any $a$.
Unfortunately, even if you assume that $\pi$ is projective and $E$ is the entire exceptional set(!), it is possible that this fails. The condition you need is that $-E$ has to be relatively ample for $\pi$. This can fail already for surfaces. For instance, assume that $E$ has two components, both with negative self-intersection, say $-n$ and $-m$, and let's say that the intersection number of the two components is $n+e$ for some positive number $e$. The intersection matrix has to be negative definite, which means that we need $nm> (n+e)^2$, but this is easy to satisfy by making $m$ really big. In this case, $E$ restricted to the component with self-intersection $-n$ has positive degree, so $-E$ cannot be ample.
This suggests that you cannot allow $E$ to have more than one component. On the other hand, in that case you are OK. Alternatively, if you allow different coefficients for the components of $E$ then a similar statement holds.