The agricultural sector is rapidly evolving to meet growing global food demands, yet tasks like fruit and vegetable handling remain labor-intensive, causing inefficiencies and post-harvest losses. Automation, particularly selective harvesting, offers a viable solution, with soft robotics emerging as a key enabler. This study introduces a novel hybrid gripper for tomato harvesting, incorporating a rigid outer frame with a soft auxetic internal lattice. The six-finger, 3D caging-effect design enables gentle yet secure grasping in unstructured environments. Uniquely, the work investigates the effect of auxetic lattice orientation on grasping conformability, combining experimental validation with 2D Digital Image Correlation (DIC) and nonlinear finite element analysis (FEA). Auxetic configurations with unit cell inclinations of 0 deg, 30 deg, 45 deg, and 60 deg are evaluated, and their grasping forces, deformation responses, and motor torque requirements are systematically compared. Results demonstrate that lattice orientation strongly influences compliance, contact forces, and energy efficiency, with distinct advantages across configurations. This comparative framework highlights the novelty of tailoring auxetic geometries to optimize robotic gripper performance. The findings provide new insights into soft-rigid hybrid gripper design, advancing automation strategies for precision agriculture while minimizing crop damage.