This work deals with scattering by obstacles which are finite disjoint unions of strictly convex bodies with smooth boundaries in an odd dimensional Euclidean space. The class of obstacles of this type is considered which are contained in a given (large) ball and have some additional properties: its connected components have bounded eccentricity, the distances between different connected components are bounded from below, and a uniform ‘no eclipse condition’ is satisfied. It is shown that if an obstacle K in this class has connected components of sufficiently small diameters, then there exists a horizontal strip near the real axis in the complex upper half-plane containing infinitely many scattering resonances (poles of the scattering matrix), i.e. the Modified Lax-Phillips Conjecture holds for such K. This generalizes a well-known result of M. Ikawa concerning balls with the same sufficiently small radius.