This paper is concerned with a generalization of the almost disturbance decoupling problem by state feedback. Apart from approximate decoupling from the external disturbances to a first to-be-controlled output, we require a second output to be uniformly bounded with respect to the accuracy of decoupling. The problem is studied using the geometric approach to linear systems. We introduce some new almost controlled invariant subspaces and study their geometric structure. Necessary and sufficient conditions for the solvability of the above problem are formulated in terms of these controlled invariant subspaces. A conceptual algorithm is introduced to calculate the feedback laws needed to achieve the design purpose.