Finding an occluded object in a lateral access environment such as a shelf or cabinet is a problem that arises in many contexts such as warehouses, retail, healthcare, shipping, and homes. While this problem, known as mechanical search, is well-studied in overhead access environments, lateral access environments introduce constraints on the poses of objects and on available grasp actions, and pushing actions are preferred to preserve the environment structure. We propose LAXRAY (Lateral Access maXimal Reduction in support Area of occupancY distribution): a system that combines target object occupancy distribution prediction with a mechanical search policy that sequentially pushes occluding objects to reveal a given target object. For scenarios with extruded polygonal objects, we introduce two lateral-access search policies that encode a history of predicted target distributions and can plan up to three actions into the future. We introduce a First-Order Shelf Simulator (FOSS) and use it to evaluate these policies in 800 simulated random shelf environments per policy. We also evaluate in 5 physical shelf environments using a Fetch robot with an embedded PrimeSense RGBD Camera and an attached pushing blade. The policies outperform baselines by up to 25 % in simulation and up to 60% in physical experiments. Additionally, the two-step prediction policy is the highest performing in simulation for 8 objects with a 69 % success rate, suggesting a tradeoff between future information and prediction errors. Code, videos, and supplementary material can be found at https://sites.google.com/berkeley.edu/lax-ray.