Vision-only grasping systems are fundamentally constrained by calibration errors, sensor noise, and grasp pose prediction inaccuracies, leading to unavoidable contact uncertainty in the final stage of grasping. High-bandwidth tactile feedback, when paired with a well-designed tactile-reactive controller, can significantly improve robustness in the presence of perception errors. This paper contributes to controller design by proposing a purely tactile-feedback grasp-adjustment algorithm. The proposed controller requires neither prior knowledge of the object's geometry nor an accurate grasp pose, and is capable of refining a grasp even when starting from a crude, imprecise initial configuration and uncertain contact points. Through simulation studies and real-world experiments on a 15-DoF arm-hand system (featuring an 8-DoF hand) equipped with fingertip tactile sensors operating at 200 Hz, we demonstrate that our tactile-reactive grasping framework effectively improves grasp stability.