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Many drawbacks in chaos-based applications emerge from the chaotic maps' poor dynamic properties. To address this problem, in this paper a chaotification model based on modulo operator and secant functions to augment the dynamic properties of existing chaotic maps is proposed. It is demonstrated that by selecting appropriate parameters, the resulting map can achieve a higher Lyapunov exponent than its seed map. This chaotification method is applied to several well-known maps from the literature, and it produces increased chaotic behavior in all cases, as evidenced by their bifurcation and Lyapunov exponent diagrams. Furthermore, to illustrate that the proposed chaotification model can be considered in chaos-based encryption and related applications, a voice signal encryption process is considered, and different tests are being used with respect to attacks, like brute force, entropy, correlation, and histogram analysis.