Routing Problems are central to many real-world applications, yet remain challenging due to their (NP-)hard nature. Amongst existing approaches, heuristics often offer the best trade-off between quality and scalability, making them suitable for industrial use. While Reinforcement Learning (RL) offers a flexible framework for designing heuristics, its adoption over handcrafted heuristics remains incomplete. Existing learned methods still lack the ability to adapt to specific instances and fully leverage the available computational budget. Current best methods either rely on a collection of pre-trained policies, or on RL fine-tuning; hence failing to fully utilize newly available information within the constraints of the budget. In response, we present MEMENTO, an approach that leverages memory to improve the search of neural solvers at inference. MEMENTO leverages online data collected across repeated attempts to dynamically adjust the action distribution based on the outcome of previous decisions. We validate its effectiveness on the Traveling Salesman and Capacitated Vehicle Routing problems, demonstrating its superiority over tree-search and policy-gradient fine-tuning; and showing that it can be zero-shot combined with diversity-based solvers. We successfully train all RL auto-regressive solvers on large instances, and verify MEMENTO's scalability and data-efficiency: pushing the state-of-the-art on 11 out of 12 evaluated tasks.