Six non-affine 3-parameter families of complex Hadamard matrices of order 8 are presented. These families contain Hadamard matrices that are not equivalent to any previously known Hadamard matrices in the literature. Each family arises from unimodular points of an affine variety defined by palindromic polynomials. The families are given as an image of a function that solves the corresponding system of polynomials on a domain that guarantees unimodularity of the solutions