AlphaGo Zero pioneered the concept of two-head neural networks in Monte Carlo Tree Search (MCTS), where the policy output is used for prior action probability and the state-value estimate is used for leaf node evaluation. We propose a three-head neural net architecture with policy, state- and action-value outputs, which could lead to more efficient MCTS since neural leaf estimate can still be back-propagated in tree with delayed node expansion and evaluation. To effectively train the newly introduced action-value head on the same game dataset as for two-head nets, we exploit the optimal relations between parent and children nodes for data augmentation and regularization. In our experiments for the game of Hex, the action-value head learning achieves similar error as the state-value prediction of a two-head architecture. The resulting neural net models are then combined with the same Policy Value MCTS (PV-MCTS) implementation. We show that, due to more efficient use of neural net evaluations, PV-MCTS with three-head neural nets consistently performs better than the two-head ones, significantly outplaying the state-of-the-art player MoHex-CNN.