We proposed, in a recent paper (10.1002/nme.5987), a fast 3D parallel Delaunay kernel for tetrahedral mesh generation. This kernel was however incomplete in the sense that it lacked the necessary mesh improvement tools. The present paper builds on that previous work and proposes a fast parallel mesh improvement stage that delivers high-quality tetrahedral meshes compared to alternative open-source mesh generators. Our mesh improvement toolkit includes edge removal and improved Laplacian smoothing as well as a brand new operator called the Growing SPR Cavity, which can be regarded as the mother of all flips. The paper describes the workflow of the new mesh improvement schedule, as well as the details of the implementation. The result of this research is a series of open-source scalable software components, called HXT, whose overall efficiency is demonstrated on practical examples by means of a detailed comparative benchmark with two open-source mesh generators: Gmsh and TetGen.