I am just posting my answer as a comment. There is a philosophical point here. Brill-Noether theory describes all linear systems on a generic curve $C$ of genus $g$. However, a curve $C$ of large genus $g$ that is a member of a pencil of curves in a surface is special by the theorem of Harris-Mumford-Eisenbud: for $g\geq 24$, a generic curve $C$ of genus $g$ is not a member of a pencil of curves on a surface -- in fact (every desingularization of every projective model) of the moduli space of genus $g$ curves is of general type. So curve $C$ that can we study as moving divisors on a surface need not be Brill-Noether general. Having said that, there is a beautiful theorem of Lazarsfeld that for a polarized K3 surface of Picard rank $1$, a general smooth curve $C$ in the complete linear system of the primitive polarizing class is Brill-Noether general.

Let $e\geq 2$ be an integer, and let $f:X\to\mathbb{P}^2$ be a degree $2$ cover branched over a smooth curve $B$ of degree $2e$. Then $f^\# :\mathcal{O}_{\mathbb{P}^2}\to f_*\mathcal{O}_X$ has quotient equal to the invertible sheaf $\mathcal{O}_{\mathbb{P}^2}(-e)$. By the computation of cohomology of invertible sheaves on projective space, the Ext group is zero, so $f_*\mathcal{O}_X$ is isomorphic to $\mathcal{O}_{\mathbb{P}^2}\oplus \mathcal{O}_{\mathbb{P}^2}(-e)$. Thus, $f_*(f^*\mathcal{O}_{\mathbb{P}^2}(d))$ is isomorphic to $\mathcal{O}_{\mathbb{P}^2}(d)\oplus \mathcal{O}_{\mathbb{P}^2}(d-e)$. Thus, for $d=1$ and $e\geq 2$, every smooth member $C$ of the basepoint free, complete linear system of the ample invertible sheaf $f^*\mathcal{O}_{\mathbb{P}^2}(1)$ is a hyperelliptic curve of genus $g=e-1$. Thus, for every odd integer $d\leq 2g-3 = 2e-5$, for every $r\geq 1$, every $\mathfrak{g}^r_d$ on $C$ has a basepoint.

I am just posting my comment as an answer. There is a philosophical point here. Brill-Noether theory describes all linear systems on a generic curve $C$ of genus $g$. However, a curve $C$ of large genus $g$ that is a member of a pencil of curves in a surface is special by the theorem of Harris-Mumford-Eisenbud: for $g\geq 24$, a generic curve $C$ of genus $g$ is not a member of a pencil of curves on a surface -- in fact (every desingularization of every projective model) of the moduli space of genus $g$ curves is of general type. So curve $C$ that can we study as moving divisors on a surface need not be Brill-Noether general. Having said that, there is a beautiful theorem of Lazarsfeld that for a polarized K3 surface of Picard rank $1$, a general smooth curve $C$ in the complete linear system of the primitive polarizing class is Brill-Noether general.

Let $e\geq 2$ be an integer, and let $f:X\to\mathbb{P}^2$ be a degree $2$ cover branched over a smooth curve $B$ of degree $2e$. Then $f^\# :\mathcal{O}_{\mathbb{P}^2}\to f_*\mathcal{O}_X$ has quotient equal to the invertible sheaf $\mathcal{O}_{\mathbb{P}^2}(-e)$. By the computation of cohomology of invertible sheaves on projective space, the Ext group is zero, so $f_*\mathcal{O}_X$ is isomorphic to $\mathcal{O}_{\mathbb{P}^2}\oplus \mathcal{O}_{\mathbb{P}^2}(-e)$. Thus, $f_*(f^*\mathcal{O}_{\mathbb{P}^2}(d))$ is isomorphic to $\mathcal{O}_{\mathbb{P}^2}(d)\oplus \mathcal{O}_{\mathbb{P}^2}(d-e)$. Thus, for $d=1$ and $e\geq 2$, every smooth member $C$ of the basepoint free, complete linear system of the ample invertible sheaf $f^*\mathcal{O}_{\mathbb{P}^2}(1)$ is a hyperelliptic curve of genus $g=e-1$. Thus, for every odd integer $d\leq 2g-3 = 2e-5$, for every $r\geq 1$, every $\mathfrak{g}^r_d$ on $C$ has a basepoint.