$\DeclareMathOperator\gon{gon}$Let $C$ be a smooth irreducible projective curve defined over complex numbers. Recall that the gonality of $C$, $\gon(C)$, is defined to be the minimal possible degree of a dominant morphism $C\to\mathbb P^1$.

I am interested in curves $C$ such that there is only one linear system $g_d^1$ satisfying $d=\gon(C)$. Examples of such curves are hyperelliptic curves of genus $\geq 2$ or trigonal curves of genus $\geq 5$, or more generally, $p$-gonal curves of genus $\geq (p-1)^2+1$ for $p$ a prime number. See, for instance, Mathew - Hyperelliptic and trigonal curves for a reference.

I am interested in the converse of the above property. To be precise,

Let $C$ be a smooth irreducible projective smooth curve defined over $\mathbb C$ such that there is only one linear system $g_d^1$ satisfying $d=\gon(C)$. What can we say aboout $d$, $g=\operatorname{genus}(C)$? Do we have any restrictions for relations of $d$ and $g$ (e.g., some nontrivial inequalities not predicted by Brill–Noether–Petri theory?)

Any comments are welcome!