  We show that the Hurwitz scheme $\mathcal{H}_{g,d}$ parametrizing $d$-sheeted
simply branched covers of the projective line by smooth curves of genus $g$, up
to isomorphism, is unirational for $(g,d)=(10,8)$ and $(13,7)$. The
unirationality is settled by using liaison constructions in $\mathbb{P}^1
\times \mathbb{P}^2$ and $\mathbb{P}^6$ respectively, and through the explicit
computation of single examples over a finite field.
