We first present a general property of nice polynomials, which leads to an important modification of the concept of equivalence classes of nice polynomials. We then establish several properties of nice symmetric and antisymmetric polynomials with an odd number of distinct roots. We give a complete description of all nice symmetric and antisymmetric polynomials with three distinct roots. We then apply our results to nice symmetric sextics and octics. As an important application, all the properties we have established have dramatically increased the speed of a computer search for examples, allowing us to discover the first examples of nice symmetric and antisymmetric polynomials with five distinct roots and the first two examples of nice polynomials with six distinct roots.