Insects and birds are often faced by opposing requirements for agile and stable flight. Here, we explore the interplay between aerodynamic effort, maneuverability, and stability in a model system that consists of a Λ-shaped flyer hovering in a vertically oscillating airflow. We determine effective conditions that lead to periodic hovering in terms of two parameters: the flyer’s shape (opening angle) and the effort (flow acceleration) needed to keep the flyer aloft. We find optimal shapes that minimize effort. We then examine hovering stability and observe a transition from unstable, yet maneuverable, to stable hovering. Interestingly, this transition occurs at post-optimal shapes, that is, at increased aerodynamic effort. These results have profound implications on the interplay between stability and maneuverability in live organisms as well as on the design of man-made air vehicles.