Calculating electronically excited states for molecular systems with an electronic configuration that has two or more unpaired electrons remains a challenge in quantum chemistry. The electronically excited states are often hard to model due to the complex nature of the wave function. This especially holds true in the case of heavy-element systems, for which the treatment of spin-orbit coupling (SOC) is crucial for a good description. To address this, we have modified the SOC-corrected restricted open-shell Kohn-Sham (ROKS) time-dependent density functional theory approach within the Tamm-Dancoff approximation (TDA). We have extended the ROKS-TDA-SOC method by including scalar relativistic spin-flip-down states (in addition to the spin-conserved and spin-flip-up ones) from ROKS-TDA, which participate in the SOC state interaction. Our assessment shows that ROKS-TDA-SOC is able to efficiently calculate the lowest-lying SOC-split excitation energies in molecular systems that contain heavy elements and two or more unpaired electrons.