Suction cups are an important gripper type in industrial robot applications, and the prior literature focuses on using vision-based planners to improve grasping success in these tasks. Vision-based planners can fail due to adversarial objects or lose generalizability for unseen scenarios, without retraining learned algorithms. In this article, we propose haptic exploration to improve suction cup grasping when visual grasp planners fail. We present the smart suction cup, an end effector that utilizes internal flow measurements for tactile sensing. We show that model-based haptic search methods, guided by these flow measurements, improve grasping success by up to 2.5× as compared with using only a vision planner during a bin-picking task. In characterizing the smart suction cup on both geometric edges and curves, we find that flow rate can accurately predict the ideal motion direction even with large postural errors. The smart suction cup includes no electronics on the cup itself, such that the design is easy to fabricate and haptic exploration does not damage the sensor. This work motivates the use of suction cups with autonomous haptic search capabilities in especially adversarial scenarios.