We improve the linear tetrahedron method to overcome systematic errors due to overestimations (underestimations) in integrals for convex (concave) functions, respectively. Our method is applicable to various types of calculations such as the total energy, the charge (spin) density, response functions, and the phonon frequency, in contrast with the Blochl correction, which is applicable to only the first two. We demonstrate the ability of our method by calculating phonons in MgB2 and fcc lithium.