See Proposition 4.5 of my paper with D. Abramovich, "Lang’s Conjectures, Fibered Powers, and Uniformity", New York J. Math. 2 (1996) 20–34.
A supersingular elliptic curve in characteristic $p>2$ will always be covered by the hyperelliptic curve $y^2=x^p-x$ as the Jacobian of the latter is isogenous to a product of supersingular elliptic curves. When $p \equiv 2 \pmod{3}$ this can be made very explicit for $y^2=x^3-1$, which is covered by $y^2=x^{p+1}-1$ in the obvious way and $y^2=x^{p+1}-1$ is isomorphic to $y^2=x^p-x$ by sending a Weierstrass point to infinity.