Let $C$ be a curve. Then I know of two ways to create morphisms. To get morphisms *from* $C$, take a line bundle of any degree $L$ and use the linear system it determines to get a map into projective space, which may or may not be injective, so you get a map to another curve. To get a morphism *to* $C$, a particular case is to take a line bundle $\mu$ of order 2, which gives us a curve $\tilde{C}$ as a double cover of $C$.

How do these two methods interact? For instance, given a curve $C$, and the tower $\tilde{C}\to C\to \bar{C}$ with the first map given as a double cover by $\mu$, and the second map given by a line bundle $L$, presumably the geometry of this tower is connected to the line bundle $L\otimes\mu$, but it's not obvious to me how to connect the two notions to get any actual information.