I was reading a proof that used the following result

Let $C$ be a hyperelliptic of genus $\ge 3$ and $\tau \colon C \to C$ the hyperelliptic involution. If $D$ is an effective divisor of degree $g-1$ such that $h^0(D)>1$ then $D = x + \tau(x) + D'$ where $D'$ is an effective divisor.

My question is, how is this result proved? It seems equivalent to showing that $|D|$ contains the unique $g^1_2$ and this made me think of Clifford's theorem but this didn't lead to much. For $g = 3$ the result holds because then $|D| = g^1_2$. But already for $g = 4$ I'm stuck. I tried playing around with the Riemann-Roch theorem but didn't get far.