We propose a framework for designing observers possessing global convergence properties and desired asymptotic behaviors for the state estimation of nonlinear systems. The proposed scheme consists in combining two given continuous-time observers: One, denoted as global, ensures (approximate) convergence of the estimation error for any initial condition ranging in some prescribed set, while the other, denoted as local, guarantees a desired local behavior. We make assumptions on the properties of these two observers, and not on their structures, and then explain how to unite them as a single scheme using hybrid techniques. Two case studies are provided to demonstrate the applicability of the framework. Finally, a numerical example is presented.