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In school choice problems, the motivation for students' welfare (efficiency) is restrained by concerns to respect schools' priorities (fairness). Among the fair matchings, even the best one in terms of welfare (SOSM) is inefficient. Moreover, any mechanism that improves welfare over the SOSM is manipulable by the students. First, we characterize the "least manipulable" mechanisms in this class: monotonically-promoting transformation proofness ensures that no student is better off by promoting their assigned school under the true preferences. Second, we use the notion that a matching is less unfair if it yields a smaller set of students whose priorities are violated, and define minimal unfairness accordingly. We then show that the Efficiency Adjusted Deferred Acceptance (EADA) mechanism is minimally unfair in the class of efficient and monotonically-promoting transformation proof mechanisms. When the objective is to improve students' welfare over the SOSM, this characterization implies an important insight into the frontier of the main axioms in school choice.