ag.algebraic geometry - Technique to prove basepoint-freeness

Let $X$ be a smooth projective variety over $mathbb{C}$.
And let $L$ be a big and nef line bundle on $X$.
I want to prove $L$ is semi-ample($L^m$ is basepoint-free for some $m > 0$).

The only way I know is using Kawamata basepoint-free theorem:

Theorem. Let $(X, Delta)$ be a proper klt pair with $Delta$ effective.
Let $D$ be a nef Cartier divisor such that $aD-K_X-Delta$ is nef and big for some
$a > 0$. Then $|bD|$ has no basepoints for all $b >> 0$.

Question. What other kinds of techniques to prove semi-ampleness or basepoint-freeness
of given line bundle are?

Maybe I miss some obvious method. Please don't hesitate adding answer although you think your idea on the top of your head is elementary.

Addition : In my situation, $X$ is a moduli space $overline{M}_{0,n}$.
In this case, Kodaira dimension is $-infty$.
More generally, I want to think genus 0 Kontsevich moduli space of stable maps to
projective space, too.
$L$ is given by a linear combination of boundary divisors.
It is well-known that boundary divisors are normal crossing,
and we know many curves on the space such that we can calculate intersection numbers with boundary divisors explicitely.