Upper Hausdorff dimension estimates for invariant sets of a class of not necessarily invertible and piecewise smooth maps on manifolds with controllable preimages of the non-differentiability sets are given in terms of the singular values of the derivative of the smoothly extended map. These estimates generalize some well-known Douady-Oesterlé type results for differentiable maps in a Euclidean space. A special class of non-injective piecewise C¹-maps is considered for which a certain factor describing the "degree of non-injectivity" is introduced. This factor is included in the dimension estimate in order to weaken the contraction condition for the singular value function of the tangent map. The contraction of the singular value function of some iteration of the map is investigated with respect to a given partition of the invariant set into "good" and "bad" parts and with respect to a decomposition of the iteration (relative to the long time behavior).