This article introduces a novel attitude controller for spacecraft subject to actuator saturation and multiple exclusion cone constraints. The proposed solution relies on a two-layer approach, where the first layer prestabilizes the system dynamics, whereas the second layer enforces constraint satisfaction by suitably manipulating the reference of the prestabilized system. In particular, constraint satisfaction is guaranteed by taking advantage of set invariance properties, whereas asymptotic convergence is achieved by implementing a nonconservative navigation field, which is devoid of undesired stagnation points. Multiple numerical examples illustrate the good behavior of the proposed scheme.