A shifting tool for use in displacing a component of a well tool can include an inner mandrel, at least one shifting key, at least one reset dog, and a retraction sleeve. The shifting key retracts relative to the inner mandrel in response to displacement of the retraction sleeve relative to the shifting key, and the reset dog extends relative to the inner mandrel in response to displacement of the inner mandrel relative to the reset dog. A method of operating a shifting tool can include engaging the shifting tool with a component of a well tool in a well, and applying a force from the shifting tool to the well tool component, thereby causing one or more reset dogs to extend into engagement with the well tool component.