In order to ensure input-output stability for plants experiencing passivity violations, this paper explores the use of the conic sector theorem for both stability analysis and controller design. Given a previously designed controller and a plant that has experienced a (partially unknown) passivity violation, a novel conic bound selection procedure is presented. This procedure can be used to assess input-output stability of the violated plant and original controller via the conic sector theorem. Should input-output stability not be ensured, two original controller synthesis methods are suggested: one is designed to mimic the controller, and the other is inspired by strictly positive real controller synthesis. Both methods guarantee input-output stability by selecting controllers within appropriate conic sectors, and involve only the evaluation of readily solvable linear matrix inequalities and algebraic Riccati equations. Along with details on computational considerations, numerical simulations are provided, demonstrating the effectiveness of the proposed methods.