Recently it has been established that given an invertible frame multiplier with semi-normalized symbol, a specific dual of any of the two involved frames can be determined for the inversion purpose. The inverse can be represented as a multiplier with the reciprocal symbol, this particular dual of one of the given frames, and any dual of the other frame. The specific dual is the only one having this property among all Bessel sequences. In this manuscript we extend the results showing that the specific dual with the above mentioned property is unique among all possible sequences. Furthermore, we allow the symbol to be not necessarily semi-normalized. Finally we characterize cases when the canonical dual frame and the new specific dual frame coincide.