This article introduces a method that generates a hexahedral-dominant mesh from an input tetrahedral mesh. It follows a three-steps pipeline similar to the one proposed by Carrier-Baudoin et al.: (1) generate a frame field; (2) generate a pointset P that is mostly organized on a regular grid locally aligned with the frame field; and (3) generate the hexahedral-dominant mesh by recombining the tetrahedra obtained from the constrained Delaunay triangulation of P. For step (1), we use a state of the art algorithm to generate a smooth frame field. For step (2), we introduce an extension of Periodic Global Parameterization to the volumetric case. As compared with other global parameterization methods (such as CubeCover), our method relaxes some global constraints and avoids creating degenerate elements, at the expense of introducing some singularities that are meshed using non-hexahedral elements. For step (3), we build on the formalism introduced by Meshkat and Talmor, fill-in a gap in their proof and provide a complete enumeration of all the possible recombinations, as well as an algorithm that efficiently detects all the matches in a tetrahedral mesh. The method is evaluated and compared with the state of the art on a database of examples with various mesh complexities, varying from academic examples to real industrial cases. Compared with the method of Carrier-Baudoin et al., the method results in better scores for classical quality criteria of hexahedral-dominant meshes (hexahedral proportion, scaled Jacobian, etc.). The method also shows better robustness than CubeCover and its derivatives when applied to complicated industrial models.