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Stacking increases storage efficiency in shelves, but the lack of visibility and accessibility makes the mechanical search problem of revealing and extracting target objects difficult for robots. In this paper, we extend the lateral-access mechanical search problem to shelves with stacked items and introduce two novel policies -- Distribution Area Reduction for Stacked Scenes (DARSS) and Monte Carlo Tree Search for Stacked Scenes (MCTSSS) -- that use destacking and restacking actions. MCTSSS improves on prior lookahead policies by considering future states after each potential action. Experiments in 1200 simulated and 18 physical trials with a Fetch robot equipped with a blade and suction cup suggest that destacking and restacking actions can reveal the target object with 82--100% success in simulation and 66--100% in physical experiments, and are critical for searching densely packed shelves. In the simulation experiments, both policies outperform a baseline and achieve similar success rates but take more steps compared with an oracle policy that has full state information. In simulation and physical experiments, DARSS outperforms MCTSSS in median number of steps to reveal the target, but MCTSSS has a higher success rate in physical experiments, suggesting robustness to perception noise. See https://sites.google.com/berkeley.edu/stax-ray for supplementary material.