This work investigates the problem of making the output of a hybrid linear system totally insensitive to a disturbance input, while attaining global asymptotic stability of the compensated hybrid dynamics. In the presence of possibly nonuniformly spaced state jumps, meeting the only condition that the minimum time between two consecutive occurrences is no smaller than a given positive real constant, both existential and constructive necessary and sufficient conditions for solvability of the problem addressed are proven.