Consider a morphism $f:\mathbb{P}^1 \to \mathbb{P}^1$ of degree $e> 2$. Using Grothendieck's splitting lemma, it is straightforward to compute that $f_*\mathcal{O}(1)$ is isomorphic to $\mathcal{O}^{\oplus 2} \oplus \mathcal{O}(-1)^{\oplus (e-2)}$. Thus the determinant is $\mathcal{O}(-(e-2))$, which is anti-ample. Thus, I don't see any reason at all that $\text{det}(f_*L)$ should be ample in general.