Let $C$ be a hyperelliptic curve of genus $g$ and let $D$ be a divisor on $C$ of degree $g+1$. Assume that the linear system $|D|$ is base-point-free. Now add a $2$-torsion point $[E]$ to $D$. I would like to know if the linear system $|D+E|$ is again base-point-free.

I think of the hyperelliptic curve as the smooth intersection of a $2$-dimensional rational normal scroll and a quadric. If it helps, I would be happy if the above statement is true for a general hyperelliptic curve in this sense, meaning that I get to choose a generic quadric.