This paper investigates the influence of adaptive walks heuristics within local searches, by studying to what extent a wiser choice among improving neighbors influences the expected quality of the attained local optima. To this aim, we specifically focus on hill-climbers and introduce the maximum expansion pivoting rule, which selects the improving neighbor having the highest number of improving neighbors. Experiments show that having one step ahead of information allows for wiser neighbor choices, leading to better local optima. As the best improvement climber, a maximum expansion climber selects a particular search trajectory among all possible first improvement trajectories, however, it significantly increases the expected quality of the trajectory. On the other hand, the computational overhead makes this heuristic less valuable when included in an iterated search process, where repeating fast random hill-climbings from random starting points still allows a better exploration of the search space. This paper, therefore, extends previous studies on the relative efficiency of hill-climbing pivoting rules, by focusing on the original maximum expansion selection criterion. Results suggest that the achievement of good local optima combined with the use of approximation techniques and problem specificities could lead to the design of effective advanced metaheuristics that exploit the maximum expansion principle.