Similarly to Tom's answer any vector bundle of rank $r$ on a variety of dimension $n<r$ contains a non-trivial subbundle. In fact it always contains a non-trivial sub line bundle (and I mean sub-bundle not sub-sheaf). The proof of this fact is a variant of the proof of Bertini's theorem. Twist your vector bundle so that it is generated by global sections and then find a section that is not zero at any point.

Similarly to Tom's answer any vector bundle of rank $r$ on a variety of dimension $n<r$ contains a non-trivial subbundle. In fact it always contains a non-trivial sub line bundle (a sub-bundle not sub-sheaf). The proof of this fact is a variant of the proof of Bertini's theorem. Twist your vector bundle so that it is generated by global sections and then find a section that is not zero at any point.