ag.algebraic geometry – Extension of a holomorphic vector bundle on a nodal curve

I am reading a paper on holomorphic curves and stuck in an argument about extension of a given holomorphic vector bundle over a nodal curve.

Let $C$ be a nodal curve without closed componets and $E$ a holomorphic vector bundle on $C$. For a compact nodal curve $tilde{C}$ containing $C$, how can $E$ extend to a holomorphic vector bundle $tilde{E}$ on $tilde{C}$? Moreover, in the same paper, the author claims that one can choose $tilde{E}$ in such a way that $langle c_{1}(tilde{E}), tilde{C}_{i} rangle$ is sufficiently large for any component $tilde{C}_{i} subset tilde{C}$. Could you please tell me how to take such an extension?

Any hint and comment are really appreciated. Thank you in advance.