Canonical perturbation theory (CPT) is a powerful tool in the field of molecular physics. It consists of a series of coordinate transformations aimed at rewriting the Hamiltonian in a simpler form without modifying the geometry of the phase space. The major achievement of CPT is the straightforward derivation of relations between the physically meaningful parameters of potential energy surfaces and the coefficients of the so-called effective Hamiltonians. While most of the studies performed up to date deal with surfaces expanded in polynomial series around a single minimum, CPT has also been applied to mixed polynomial/trigonometric expansions in the treatment of torsions. In this latter case, however, the accuracy of CPT has not been verified. The goal of this article is to suggest some modifications of the procedures, which allow for the successful application of CPT to floppy molecules with several equilibrium positions and nonpolynomial expansions. The levels belonging to all the wells or located above the saddle points are satisfactorily reproduced by the perturbative Hamiltonian. More precisely, the vibrational modes are sorted into two categories, namely oscillator-like ones and hindered-rotor-like ones. The application of CPT enables the expression of the Hamiltonian in terms of the good quantum numbers and/or classical constants of the motion associated with the oscillator-like modes. The perturbative Hamiltonian then acts on the reduced dimensional space of the hindered-rotor-like modes. The validity and accuracy of this approach are tested on two-dimensional and three-dimensional models mimicking, respectively, nonlinear and linear HCN.