Run-and-tumble is a basic model of persistent motion and a motility strategy widespread in micro-organisms and individual cells. In many natural settings, movement occurs in the presence of confinement. While accumulation at the surface has been extensively studied, the transport parallel to the boundary has received less attention. We consider a run-and-tumble particle confined inside a slit, where motion in the bulk alternates with intermittent sojourns at the wall. We first propose a discrete-direction model that is fully tractable and obtain the exact diffusion coefficient characterizing the long-time exploration of the slit. We then use numerical simulations to show that with an adequate choice of parameters, our analytical prediction provides a useful approximation for the diffusion coefficient of run-and-tumble with continuous direction. Finally, we identify the conditions that maximize diffusion within the slit and discuss the optimal mean run time. For swimming bacteria, we find that the optimum is typically reached when the mean run length is comparable to the confinement size.