**Edit:** As pointed out by Sasha, my original argument was not complete for case (2). One can argue as follows: assuming $p\leq q$, the maximum number $\ell$ of points of $C$ lying on a line is $\leq q\ $ — otherwise the line is contained in $C$.
By B. Basili, *Indice de Clifford des intersections complètes de l’espace*, Bull. S. M. F. **124**, no. 1 (1996), p. 61-95,  the gonality of $C$ is $pq-\ell$, hence $\geq p(q-1)$. But if $C$ is a 2-sheeted cover of a curve $D$ of genus $\leq 3$, a $g^{1}_3$ on $D$ lifts to a $g^1_6$ on $C$; similarly if  $C$ is a 3-sheeted cover of
an elliptic (or more generally hyperelliptic) curve $E$, a $g^1_2$ on $E$ lifts to a $g^1_6$ on $C$. Hence the gonality of $C$ is $\leq 6$, which is less than $p(q-1)$.