Accurate final grasp alignment remains challenging for edge-prominent objects such as thin plates, discs, and rods, whose sparse contacts are easily occluded and poorly resolved by depth sensing. We present TacRefineNet, a tactile-only, goal-conditioned framework for local refinement along tactilely observable pose dimensions. Given current and target multi-finger tactile images and their corresponding hand-joint configurations, a Siamese policy network directly predicts corrective wrist pose increments. The hand iteratively opens, moves, and regrasps, forming an external-dexterity tactile servoing loop. Cross-combination training pairs current and target samples, allowing targets within the sampled pose range to be specified without retraining. We collect 156,007 simulated samples from 15 plates, discs, and rods and train the policy entirely in MuJoCo before zero-shot deployment to an 11-DoF five-fingered hand with piezoresistive sensors. On seen objects, the real system achieves 80.7\% and 59.3\% success under the $10^\circ$/10\,mm criterion for fixed and random targets, respectively; after five steps, the mean errors are approximately 5.2\,mm and $3.5^\circ$. Experiments further show continuous correction under long-horizon perturbations and limited within-category transfer to unseen objects, with reduced performance for symmetric or weakly discriminative contacts. Project website is available at https://sites.google.com/view/tacrefinenet

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