Are there non-trivial $\mathbb{F}_q$-covers of the j-invariant 0 elliptic curve by a hyperelliptic or cyclic trigonal curve?

Question

Consider the ordinary elliptic curve $E\!: y^2 = x^3 + b$ (of $j$-invariant $0$) over a finite field $\mathbb{F}_q$ such that $\sqrt{b}, \sqrt[3]{b} \not\in \mathbb{F}_q$. Also, for any $n \in \mathbb{N}$ we have the hyperelliptic curve $H_n\!: y^2 = x^{3n} + b$ and the cyclic trigonal curve $T_n\!: x^3 = y^{2n} - b$. There are the obvious covers $$ H_n \to E \qquad (x,y) \mapsto (x^{n}, y), \qquad\qquad T_n \to E \qquad (x,y) \mapsto (x, y^{n}) $$ Are there other $\mathbb{F}_q$-covers of $E$ by a hyperelliptic or cyclic trigonal curve?

Thanks in advance.