Input-output stability theory is an indispensable tool in control engineering. However, classic results such as the Conic Sector Theorem, cannot be applied in some important cases. This technical note extends the original Conic Sector Theorem to apply to open-loop unstable systems and those with negative upper conic bounds and incorporates a careful treatment of degenerate conic systems and alterations accounting for nonzero initial conditions. Simulations highlight how the contributions facilitate robust control in the most challenging of circumstances.