In this paper we introduce the notion of eventually periodic switching signal. We prove that if a family of linear vector fields satisfies a mild finite time controllability condition, and for each initial state there exists a time-dependent switching signal which asymptotically drives the system to the origin (possibly allowing different signals for different initial states), then the same goal can be achieved by means of an eventually periodic switching signal. This enables us to considerably reduce the dependence of the control law on the initial state. In this sense, the problem addressed in this paper can be reviewed as a switched system theory version of the classical problem of investigating whether, or to what extent, a nonlinear asymptotically controllable system admits stabilizing feedback laws.