Pencils of small degree on curves on unnodal Enriques surfaces

We use vector-bundle techniques in order to compute $\dim W_d^1(C)$ where $C$ is general and smooth in a linear system on an unnodal Enriques surface. We furthermore find new examples of smooth curves on Enriques surfaces with an infinite number of $g_{\text{gon}(C)}^1$'s.