Let's us say we have a polystable vector bundle $V=W_1\oplus W_2$, where $W_i$'s are stable sub-bundle of degree $0$.
If $W_1\ncong W_2$, then I could easily prove that any stable bundle of $V$ degree $0$ is either contained in either $W_1$ or $W_2$ in the following manner-
If it is a line sub-bundle $L$ then obviously it is contained in either $W_1$ or $W_2$.
If we have a rank 2 degree 0 stable sub-bundle $W$ of $V$. Then we have the following maps $W\rightarrow W_1\oplus W_2\rightarrow W_1$ and $W\rightarrow W_1\oplus W_2\rightarrow W_2$. One of these maps is obviously non-zero and hence isomorphism. Therefore, $W\cong W_1$ or $W\cong W_2$ and hence contained in either either $W_1$ or $W_2$. ($\textbf{Question}$ Can we really conclude that $W$ is either contained in $W_1$ or $W_2$?)
If we have a rank 3 degree 0 stable sub-bundle $W$ of $V$. In a similar manner one can show $W$ is isomorphic to either $W_1$ or $W_2$, which is not possible, hence there is no stable sub-bundle of degree $0$ of rank 3.
Therefore, any stable bundle of $V$ degree $0$ is either contained in either $W_1$ or $W_2$.
If $W_1\cong W_2$, i.e., $V=W\oplus W$ where $W$ is stable bundle of degree $0$. $\textbf{Is it still true that any stable sub-bundle of degree 0 is contained in $W$?}$