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

The unirationality of the Hurwitz schemes $\mathcal{H}_{10,8}$ and $\mathcal{H}_{13,7}$