 Upper degree bound for line subbundles on a smooth curve

Choose a rational basis of the dual of a vector bundle and an effective divisor $D$ bounding all its poles. Evaluation gives a generically injective map $E\to {\mathcal{O}}_C(D{)}^{\oplus r}$. A line subbundle maps nontrivially to at least one summand, so ${\mathcal{O}}_C(D)\otimes L^{-1}$ has a nonzero regular section. Consequently $\mathrm{deg}L\le \mathrm{deg}D$. This bound depends on $E$ and the chosen rational frame, and does not require $E$ to be stable.