Testing the spatial variation of the fine-structure constant a indicated by Webb et al. [J. K. Webb, J. A. King, M. T. Murphy, V. V. Flambaum, R. F. Carswell, and M. B. Bainbridge, Phys. Rev. Lett. 107, 191101 (2011)] with terrestrial laboratory atomic measurements requires at least. alpha/alpha similar to 10(-19) yr(-1) sensitivity. We conduct a systematic search of atomic systems for such a test that have all features of the best optical clock transitions leading to the possibility of the frequency measurements with fractional accuracy on the level of 10(-18) or better and have a factor of 100 extra enhancement of alpha variation in comparison to experimental frequency ratio measurement accuracy. We identify the pair of actinide Cf15+ and Es16+ ions as the best system for a test of spatial alpha-variation hypothesis as it satisfies both of these requirements and has sufficiently simple electronic structure to allow for high-precision predictions of all atomic properties required for rapid experimental progress.