The synthesis of time-optimal feedback control of a single input, continuous time, linear time invariant system with bounded inputs is considered. Unlike a recent paper by the authors, the target final state is not necessarily the origin of the state space. Semi-algebraic representations of switching surfaces corresponding to the bang-bang time-optimal control are computed using a Gröbner basis based elimination algorithm. These surfaces are then used to synthesize a nested switching logic for time-optimal feedback control. Non-origin target points introduce unavoidable limit cycles in time optimal trajectories, whose period depends on the target position. This dependence is algebraically characterized and a method to compute the periods of the limit cycles is proposed. As a natural extension, we also provide a semi-algebraic characterization of the set of all points reachable with constrained inputs.