In the current work we apply the completeness-optimization paradigm [P. Manninen and J. Vaara, J. Comput. Chem. 27, 434 (2006)] to investigate the basis set convergence of the moments of the ground-state electron momentum density at the self-consistent field level of theory. We present a black-box completeness-optimization algorithm that can be used to generate computationally efficient basis sets for computing any property at any level of theory. We show that the complete basis set (CBS) limit of the moments of the electron momentum density can be reached more cost effectively using completeness-optimized basis sets than using conventional, energy-optimized Gaussian basis sets. By using the established CBS limits, we generate a series of smaller basis sets which can be used to systematically approach the CBS and to perform calculations on larger, experimentally interesting systems.