We give a supplement to the Smith reduction theorem for nonautonomous ordinary differential equations (ODEs) that satisfy the squeezing property in the case when the right-hand side is almost periodic in time. The reduction theorem states that some set of nice solutions (including the bounded ones) of a given nonautonomous ODE satisfying the squeezing property with respect to some quadratic form can be mapped one-to-one onto the set of solutions of a certain system in the space of lower dimensions (the dimensions depend on the spectrum of the quadratic form). Thus, some properties of bounded solutions to the original equation can be studied through this projected equation. The main result of the present paper is that the projected system is almost periodic provided that the original differential equation is almost periodic and the inclusion for frequency modules of their right-hand sides holds (however, the right-hand sides must be of a special type). From such an improvement we derive an extension of Cartwright's result on the frequency spectrum of almost periodic solutions and obtain some theorems on the existence of almost periodic solutions based on low-dimensional analogs in dimensions 2 and 3. The latter results require an additional hypothesis about the positive uniformly Lyapunov stability and, since we are interested in nonlinear phenomena, our existence theorems cannot be directly applied. On the other hand, our results may be applicable to study the question of sensitive dependence on initial conditions in an almost periodic system with a strange nonchaotic attractor. We discuss how to apply this kind of results to the Chua system with an almost periodic perturbation. In such a system the appearance of regular almost periodic oscillations as well as strange nonchaotic and chaotic attractors is possible.