In this paper, we propose an efficient contact planner for quadrupedal robots to navigate in extremely constrained environments such as stepping stones. The main difficulty in this setting stems from the mixed nature of the problem, namely discrete search over the steppable patches and continuous trajectory optimization. To speed up the discrete search, we study the properties of the transitions from one contact mode to another. In particular, we propose to learn a dynamic feasibility classifier and a target adjustment network. The former predicts if a contact transition between two contact modes is dynamically feasible. The latter is trained to compensate for misalignment in reaching a desired set of contact locations, due to imperfections of the low-level control. We integrate these learned networks in a Monte Carlo Tree Search (MCTS) contact planner. Our simulation results demonstrate that training these networks with offline data significantly speeds up the online search process and improves its accuracy.