Object insertion is primarily studied using rigid robotic hands. However, these may have difficulties overcoming spatial uncertainties originating from an uncertain initial grasp. Compliant hands, on the other hand, can cope with SE(3) uncertainties and adapt to the environment upon contact. Nevertheless, contact forces may contribute additional uncertainties and lead to failure if not controlled properly. In this letter, we take inspiration from human insertion and study how haptic glances with compliant hands during contact can provide valuable information regarding object state. Using a force/torque sensor, we show that a haptic glance based on excitation of finger perturbations can provide accurate contact localization and indication of a successful insertion. With such insight, we propose an online learning scheme for general precision control of contact-rich object insertion. A deep residual Reinforcement Learning (RL) policy leverages the contact dynamics of the compliant hand to cope with SE(3) uncertainties. Several experiments of precision insertion tasks with various objects and grasp uncertainties exhibit high success rate and validate the effectiveness of the approach. .