This paper focuses on the consensus tracking control of nonlinear multi-agent systems by utilizing the quadratic inner-bounded (QIB) and the one-sided Lipschitz (OSL) conditions in the presence of input saturation constraint under a directed communication topology. A novel sector constraint for the saturation function to formulate consensus control of nonlinear systems under saturating inputs is derived. This sector condition is applied to develop a leader-following consensus of nonlinear agents. A local treatment of the consensus control, ensuring a guaranteed region of stability in the presence of input constraints is provided herein. A computationally simple convex routine-based approach is attained for extraction of the coupling weight and gain of the relative state feedback-based consensus protocol along with the enlargement of the region of stability. Unlike the conventional schemes, the proposed approach can deal with OSL nonlinear agents, effectively employs the information of communication topology in the sector condition, and can be applied to both linear and nonlinear regions of actuators. Moreover, a useful consensus approach for the Lipschitz nonlinear agents is obtained as a particular scenario of the resultant method. Numerical examples to demonstrate the applications and effectiveness of the proposed approach for the input-constrained nonlinear mobile and robotic agents are provided.