High-quality volumetric meshing from medical images is a key bottleneck for physics-based simulations in personalized medicine. For volumetric meshing of complex medical structures, recent studies have often utilized deep learning (DL)-based template deformation approaches to enable fast test-time generation with high spatial accuracy. However, these approaches still exhibit limitations, such as limited flexibility at high-curvature areas and unrealistic inter-part distances. In this study, we introduce a simple yet effective snap-and-tune strategy that sequentially applies DL and test-time optimization, which combines fast initial shape fitting with more detailed sample-specific mesh corrections. Our method provides significant improvements in both spatial accuracy and mesh quality, while being fully automated and requiring no additional training labels. Finally, we demonstrate the versatility and usefulness of our newly generated meshes via solid mechanics simulations in two different software platforms. Our code is available at https://github.com/danpak94/Deep-Cardiac-Volumetric-Mesh.