Assume $X$ is a two dimensional scheme with $Pic(X)=\mathbb{Z}$ such that every rank two locally free sheaf $\mathcal{E}$ is given by an exact sequence $0\rightarrow \mathcal{O}_X(n)\rightarrow \mathcal{E}\rightarrow\mathcal{I}_Z(m)\rightarrow 0$, with $n\geq m$ and $\mathcal{I}_Z$ is the ideal sheaf of a codimension two subscheme.
Now let $\mathcal{F}$ be another rank two locally free sheaf such that $\mathcal{E}(-1)\hookrightarrow \mathcal{F}\hookrightarrow \mathcal{E}$. (Notice that we have $0\rightarrow \mathcal{O}_X(n-1)\rightarrow \mathcal{E}(-1)\rightarrow\mathcal{I}_Z(m-1)\rightarrow 0$.)
Does this imply that we either have $0\rightarrow \mathcal{O}_X(n)\rightarrow \mathcal{F}\rightarrow\mathcal{I}_Z(m-1)\rightarrow 0$ or $0\rightarrow \mathcal{O}_X(n-1)\rightarrow \mathcal{F}\rightarrow\mathcal{I}_Z(m)\rightarrow 0$ ( if n>m)?
That is can we describe $\mathcal{F}$ as an extension only using the extensions corresponding to $\mathcal{E}$ and $\mathcal{E}(-1)$?