We prove the following results on the unique continuation problem for CR mappings between real smooth hypersurfaces in ℂn. If the CR mappingH extends holomorphically to one side of the source manifoldM near the pointp0 εM, the target manifoldM′ contains a holomorphic hypersurface σ′ throughp′0 =H(p0 (i.e.,M′ is nonminimal atp′0), andH(M) ⊄ Σ′ (forcingM to be nonminimal atp0), then the transversal component ofH is not flat atp0. Furthermore, we show that the assumption thatH extends holomorphically to one side ofM cannot be removed in general. Indeed, we give an example of a smooth CR mappingH, withM, M′ ⊂ ℂ2, real analytic and of infinite type atp0 andp′0 respectively (without being Levi flat), such thatH(M) ⊄ Σ′ but the transversal component ofH is flat atp0 (in particular,H is not real analytic!). However, we show that ifM andM′ are assumed to be real analytic, and if the sourceM is “sufficiently far from being Levi flat” in a certain sense (so as to exclude the above mentioned counterexample) then the assumption thatH extends holomorphically to one side ofM can be dropped. Also, in the general case, we prove that the rate of vanishing of the transversal component cannot be too rapid (unlessH(M) ⊂ Σ′), and we relate the possible rate of vanishing to the order of vanishing of the Levi form on a certain holomorphic submanifold ofM.