Many engineering systems require both continuous and discrete valued signals, as well as continuous and discrete time scales in their descriptions and are therefore known as hybrid systems. Recently, the concept of time-scale calculus was introduced as a unification of the theory of difference and differential equations, making it an ideal tool to analyze hybrid systems. The time-scale calculus can handle solutions of hybrid systems with infinitely many jumps. However, to date time-scale calculus restricts time to a non-empty closed subset of the real line which is not general enough to fully capture the dynamics of hybrid systems. A generalization is proposed to extend time-scale calculus such that the time can be defined on more suitable subsets of the real line. Stability definitions of hybrid systems using this framework are also introduced together with a Lyapunov characterization of global uniform stability. A few examples demonstrate the usefulness of this new framework.