I will fix notation: $\Delta = \mathrm{Spec} R$ denotes a discrete valuation ring and $\Delta^*=\mathrm{Spec} K$ for $K=\mathrm{Frac}(R)$. Suppose we are given a curve $\pi:C\to \Delta$ and a line bundle $\mathcal{L}$ on $C$. Let $U = \pi^{-1}(\Delta^*)$. If I am given a section $s \in \mathcal{L}(U)$, are there any results that help me to extend this $s$ over the central fibre $C_0$ (the fibre over the closed point of $\Delta$)?

If I assume that the generic fibre is irreducible, the whole family is irreducible and our section $s$ defines a rational section $\mathcal{L}$. However, I would like to obtain a value for every point in the central fibre, possibly infinite, if it is possible. For instance, if we regard a meromorphic/rational section of $\mathcal{L}$ as being a section of $\Bbb{P}(\mathcal{L}\oplus \mathcal{O}_C)$, then we can extend the section over codimension $1$ points by properness, but this only extends $s$ to the generic point of the central fibre.

Briefly, what tools, if any, are available for extending a $s\in\mathcal{L}(U)$ to the central fibre, where we view $s$ as a section of $\Bbb{P}(\mathcal{L}\oplus \mathcal{O}_C)$?