The recently proposed extension of the method of moments of coupled-cluster equations (MMCC) to excited states via the equation-of-motion coupled-cluster (EOMCC) formalism [K. Kowalski and P. Piecuch, J. Chem. Phys. 115, 2966 (2001)] is developed further. A new approximate variant of the excited-state MMCC theory, termed the MMCC(2,4) method, is proposed and tested. In the MMCC(2,4) method, relatively simple noniterative corrections due to triples and quadruples are added to the excited-state energies obtained in the standard EOMCCSD (EOMCC singles and doubles) calculations. The performance of the MMCC(2,4) approach is illustrated by the results of calculations for the excited states of N-2, C-2, and CH+. The MMCC(2,4) energies are compared with the results of the MMCC(2,3) calculations, in which noniterative corrections due to triples only are added to the EOMCCSD energies, and with the results of other EOMCC calculations, including various EOMCC triples schemes.