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Edit: As pointed out by Sasha, my original argument was not complete for case (2). One can argue as follows: suppose $C$ is a $(p,q)$ complete intersection, with $p\geq q$. The maximum number $\ell$ of points of $C$ lying on a line is $\leq p\ $ — 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 $p(q-1)$ if $q\geq 4$.