In this paper, we present a nonlinear correction finite volume scheme that preserves the discrete extremum principle for convection-diffusion equations on tetrahedral meshes. The approximation of the diffusive flux is based on a nonlinear correction of the second-order linear flux, while the convection flux is approximated by the upwind scheme with second-order accuracy given by a Taylor expansion. In the construction of the new scheme, the requirement to represent auxiliary unknowns as convex combinations of primary unknowns can be removed, which greatly reduces the restrictions on diffusion coefficients and mesh geometry. Moreover, the obtained new scheme is cell-centered and satisfies local conservation. In addition, we theoretically testify that the numerical solution of the new scheme keeps the discrete extremum principle and verify it numerically. Numerical results also show that our scheme has almost second-order accuracy.