Subsheaves of locally free sheaves on a rational curve

Question

Let $k$ be an algebraically closed field, $\mathcal{E}$ a locally free sheaf on $\mathbb{P}^1_k$ and $\mathcal{L} \subset \mathcal{E}$ a subsheaf of rank $1$. By Grothendieck's theorem, we know that $\mathcal{E}$ (resp. $\mathcal{L}$) is of the form $\oplus_{i=1}^{\mathrm{rk}(\mathcal{E})} \mathcal{O}_{\mathbb{P}^1_k}(a_i)$ (resp. $\mathcal{O}_{\mathbb{P}^1_k}(a)$). Does there exist an injective morphism $i:\mathcal{L} \to \mathcal{E}$ such that $\mathcal{L}$ maps to just one component $\mathcal{O}_{\mathbb{P}^1_k}(a_j)$ of $\mathcal{E}$ i.e., $i$ factors through an injective morphism $i':\mathcal{L} \to \mathcal{O}_{\mathbb{P}^1_k}(a_j)$ for some $1 \le j \le \mathrm{rk}(\mathcal{E})$?