Using the linear $T$-dual Ward identity associated with the NS-NS B-field or two Neveu-Schwarz B-fields (NSNS) gauge transformations, some Ramond-Ramond (RR) couplings on ${\mathrm{D}}_{p}$-branes have been found at order $O({\ensuremath{\alpha}}^{\ensuremath{'}2})$. We examine the ${C}^{(p\ensuremath{-}1)}$ couplings with the $S$-matrix elements of one RR, one graviton and one antisymmetric B-field vertex operator. We find the consistency of $T$-dual $S$-matrix elements and explicit results of scattering string amplitude and show that the string amplitude reproduces these couplings as well as some other couplings. This illustration is found for ${C}^{(p\ensuremath{-}3)}$ couplings in the literature, which is extended to the ${C}^{(p\ensuremath{-}1)}$ couplings in this paper.