  This paper is a sequel of arXiv:2208.00990. Let $C$ be a smooth complex
projective curve of genus $g$ and let $C^{(k)}$ be its $k$-fold symmetric
product. The covering gonality of $C^{(k)}$ is the least gonality of an
irreducible curve $E\subset C^{(k)}$ passing through a general point of
$C^{(k)}$. It follows from previous works of the authors that if $2\leq k\leq
4$ and $g\geq k+4$, the covering gonality of $C^{(k)}$ equals the gonality of
$C$. In this paper, we prove that under mild assumptions of generality on $C$,
the only curves $E\subset C^{(k)}$ computing the covering gonality of $C^{(k)}$
are copies of $C$ of the form $C+p$, for some point $p\in C^{(k-1)}$. As a
byproduct, we deduce that the connecting gonality of $C^{(k)}$ (i.e. the least
gonality of an irreducible curve $E\subset C^{(k)}$ connecting two general
points of $C^{(k)}$) is strictly larger than the covering gonality.
