Remark. Note that $\operatorname{\mathbf{Hom}}(C,\mathbb P^1)$ is an open subscheme of $\operatorname{\mathbf{Div}}_{C \times \mathbb P^1}$, whereas $\operatorname{Sym}^d(C)$ is a component of $\operatorname{\mathbf{Div}}_C$. Specifically, a homomorphism $\phi \colon C \to \mathbb P^1$ corresponds to its graph, which is a section of a line bundle $\mathscr M$ on $C \times \mathbb P^1$. Since
\begin{align*}
\operatorname{\mathbf{Pic}}_{C \times \mathbb P^1} &= \operatorname{\mathbf{Pic}}_C \times \operatorname{\mathbf{Pic}}_{\mathbb P^1} \times \operatorname{\mathbf{Hom}}(\operatorname{\mathbf{Alb}}_C,\operatorname{\mathbf{Pic}}^0_{\mathbb P^1})\\
&= \operatorname{\mathbf{Pic}}_C \times \operatorname{\mathbf{Pic}}_{\mathbb P^1},
\end{align*}
the line bundle $\mathscr M$ is of the form $\mathscr L \boxtimes \mathcal O(n)$ for some line bundle $\mathscr L$ on $C$ and some $n \in \mathbb Z$. Counting intersections with $\{c\} \times \mathbb P^1$, we see that $n = 1$. Restricting to $C \times \{\sigma\}$, we see that $\mathscr L = \phi^* \mathcal O(1)$. Thus, $\phi$ corresponds to a section $s$ of $\mathscr L \boxtimes \mathcal O(1)$.
Conversely, a section $s$ of $\mathscr L \boxtimes \mathcal O(1)$ comes from a morphism $\phi$ if and only if $\mathscr L$ is effective and $V(s)$ is irreducible. Indeed, since $\mathscr L \boxtimes \mathcal O(1)$ is primitive (not divisible), $V(s)$ is reduced. Hence, the dominant morphism $V(s) \to C$ is flat, since $V(s)$ is torsion-free over $C$ since it is integral. Since it is of degree $1$, it is an isomorphism.
The only other possibility for a section $s$ of $\mathscr L \boxtimes \mathcal O(1)$ is that it contains a vertical component. Indeed, the only decomposition of $\mathscr L \boxtimes \mathcal O(1)$ as a sum of effective divisors must have $\mathscr L' \boxtimes \mathcal O$ as one of its summands, all of whose sections are vertical.
The morphism $\phi \colon C \to \mathbb P^1$ and the pair $(\mathscr L, s)$ determine each other uniquely, where two pairs $(\mathscr L, s)$, $(\mathscr L', s')$ correspond if and only if there exists an isomorphism $\alpha \colon \mathscr L \stackrel \sim \to \mathscr L'$ taking $s$ to $s'$.
Remark. Thus, the map you give can be extended to the rational map
\begin{align*}
\sigma^* \colon \operatorname{\mathbf{Div}}_{C \times \mathbb P^1} &\dashrightarrow \operatorname{\mathbf{Div}}_C \\
Z &\mapsto Z \cap (C \times \{\sigma\}),
\end{align*}
which is defined away from the divisors containing $C \times \{\sigma\}$. We will focus on the locus $U$ of $\operatorname{\mathbf{Div}}_{C \times \mathbb P^1}$ of irreducible divisors in $\mathscr L \boxtimes \mathcal O(1)$ for some $\mathscr L$ of degree $d > 2g$. We get a commutative square
$$\begin{array}{ccc} \operatorname{\mathbf{Div}}_{C \times \mathbb P^1} & \stackrel{\sigma^*}\dashrightarrow & \operatorname{\mathbf{Div}}_C \\ \downarrow & & \downarrow \\ \operatorname{\mathbf{Pic}}_{C \times \mathbb P^1} & \stackrel{\sigma^*}\rightarrow &\ \operatorname{\mathbf{Pic}}_C. \end{array}$$
Over the locus of $\operatorname{\mathbf{Pic}}_C$ where $\mathscr L$ is of degree $d > 2g$, the right vertical arrow is a $\mathbb P^r$-bundle, where $r = h^0(\mathscr L) - 1 = d - g$ (which does not depend on $\mathscr L$ but only on $d$). Similarly, over the same locus, the left vertical arrow is a $\mathbb P^{r'}$-bundle, where $r' = h^0(\mathscr L \boxtimes \mathcal O(1)) - 1$. By Künneth we have $r' = 2(r+1) - 1 = 2d - 2g + 1$.
Restricting the left hand side to the locus of the form $\mathscr L \boxtimes \mathcal O(1)$, the bottom map becomes an isomorphism. On the locus $U$ of irreducible divisors in $\mathscr L \boxtimes \mathcal O(1)$, the top map on each fibre is a morphism from an open in $\mathbb P^{2d-2g+1}$ to $\mathbb P^{d-g}$.
Remark. To describe the map more explicitly, let $[x:y]$ be coordinates on $\mathbb P^1$. For simplicity, assume $\sigma = [0:1]$. Then $H^0(\mathscr L \boxtimes \mathcal O(1))$ is given by $\{\lambda x + \mu y\ |\ \lambda,\mu \in H^0(\mathscr L)\}$, and a section $\lambda x + \mu y$ is mapped to the section $\mu \in H^0(\mathscr L)$. In particular, this is a linear map, so the fibres are linear spaces.
The locus of divisors $Z$ not containing $C \times \{0\}$ corresponds to the $\lambda x + \mu y$ with $\mu \neq 0$. This is the maximal set where the map on the projective spaces is defined, and there the fibres are $\mathbb A^{d-g+1}$ corresponding to the affine space of values of $\lambda$.
We are further restricting to divisors $Z$ not containing any vertical component; this is probably equivalent to linear independence of $\lambda$ and $\mu$, so this removes an $\mathbb A^1\setminus\{0\}$ from each fibre (corresponding to $\lambda = c \mu$). More canonically, the conditions that $\lambda$ and $\mu$ are linearly independent are cut out in $\mathbb P^{2d-2g+1}$ by the nonvanishing of certain minors.
Any further properties you want to deduce should follow from this description.
