Low-bit representation based motion estimation approaches such as one-bit transform, multiplication-free one-bit transform, and two-bit transform use matching criteria suitable for hardware. Thus, these techniques are more appropriate for consumer electronics devices which include hardware implementations in order to meet various constraints such as low-power and limited processing capability. The recently proposed constrained one-bit transform based motion estimation approach provides better results compared to other low-bit representation based approaches. Furthermore, it is shown in the literature that the performance of such approaches can be improved by employing some additional searches in the image domain starting from the best motion vectors found in the low-bit representation based motion estimation stage. This idea is applied to the constrained one-bit transform based motion estimation approach in this paper. Furthermore, a faster version of the constrained one-bit transform is also evaluated within this modification concept. Our experiments show that one of the proposed methods provides the best performance among methods sharing similar modification approaches whereas the other one enables the fastest implementation with small performance degradation.