Using the e(+) e(-) collision data collected with the BESIII detector operating at the BEPCII collider, at center-of-mass energies from the threshold to 4.95 GeV, we present precise measurements of the cross section for the process e(+) e(-) -> D-s(+) D-s(-) using a single-tag method. The resulting cross section line shape exhibits several new structures, thereby offering an input for a future coupled-channel analysis and model tests, which are critical to understand vector charmonium-like states with masses between 4 and 5 GeV.