Xiao et al. (2012) propose three-level screening designs using conference matrices. These designs require one more than twice as many runs as there are factors, and provide unbiased estimates of linear main effect in the presence of quadratic effects and two-factor interactions. However, unbiased estimates of two-factor interactions and quadratic effects can not be obtained. The article gives additional runs which provide unbiased estimates of two-factor interactions between a particular factor and other m-1 factors and quadratic effects of the factor. Furthermore, it is shown that existence of unbiased estimates ensures unbiasedness of least square estimates. Numerical experiment is investigated to illustrate efficiencies of these additional runs.