They can still give you (non-canonical) rational maps to mathbbPn:
Even when an invertible sheaf L on X has no global sections, one can still find open subsets U of X such that L|U is globally generated, for example when U is affine. This isn't so interesting, but if you can find such a U that is not contained in an affine, then L|U might not be trivial, and then you might get morphisms from U to mathbbPn (by choosing generators) not coming from mathcalOU. If X was integral, then U will be automatically dense, so you get a rational map from X to mathbbPn.
One way you could look for such a U is to pick n elements of the stalk Lx at a point x, then intersect n neighborhoods on which these elements extend, remove their common zero locus, and let the result be
U. If U isn't contained in an affine, maybe you've found something cool. If you really wanted you could try working out a nice description for the rational map you've defined (though I've never done this). Even if U was affine, maybe you've found a more interesting description of a less interesting map.
From a very different perspective, since mathcalO(−d) is the inverse of mathcalO(d), in some sense any "information" it contains is obtainable from inverting the transition functions of mathcalO(d)... so with this restrictive view of "information", perhaps it would be interesting to study what sort of morphisms/maps to mathbbPn one can get from an invertible sheaf L on X when neither L nor Lvee is ample/very ample.
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