One can do the following. Take a destabilizing $F \subset E^*$. First replace $F$ with the saturation $F'$ of $F$, i.e. the sheaf whose sections consist of those sections of $E^*$ that generically lie in $F$. There is a natural map $F \to F'$ whose cokernel is supported in codimension $1$. It follows that $\alpha_d(F')=\alpha_d(F)$ and $\alpha_{d-1}(F') \geq \alpha_{d-1}(F)$. So if $F$ is destabilizing then $F'$ is as well.

Now since $F'$ is saturated, $F'$ and $E_* /F'$ are locally free in codimension $1$. Since $\alpha_d$ and $\alpha_{d-1}$ may be calculated by ignoring any codimension $2$ locus, you can ignore the locus where they are not locally free and then use the argument for sub-vector bundles.

One can do the following. Take a destabilizing $F \subset E^*$. First replace $F$ with the saturation $F'$ of $F$, i.e. the sheaf whose sections consist of those sections of $E^*$ that generically lie in $F$. There is a natural map $F \to F'$ whose cokernel is supported in codimension $1$. It follows that $\alpha_d(F')=\alpha_d(F)$ and $\alpha_{d-1}(F') \geq \alpha_{d-1}(F)$. So if $F$ is destabilizing then $F'$ is as well.

Now since $F'$ is saturated, $F'$ and $E^* /F'$ are locally free in codimension $1$. Since $\alpha_d$ and $\alpha_{d-1}$ may be calculated by ignoring any codimension $2$ locus, you can ignore the locus where they are not locally free and then use the argument for sub-vector bundles.