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The extremely neutron-rich system H-7 was studied in the direct H-2(He-8, He-3) H-7 transfer reaction with a 26 A MeV secondary He-8 beam [Bezbakh et al., Phys. Rev. Lett. 124, 022502 (2020)]. The missing mass spectrum and center-of-mass angular distributions of H-7, as well as the momentum distribution of the H-3 fragment in the H-7 frame, were constructed. In addition, we carried out another experiment with the same beam but a modified setup, which was cross-checked by the study of the H-2(Be-10, He-3) Li-9 reaction. A solid experimental evidence is provided that two resonant states of H-7 are located in its spectrum at 2.2(5) and 5.5(3) MeV relative to the H-3 +4n decay threshold. Also, there are indications that the resonant states at 7.5(3) and 11.0(3) MeV are present in the measured H-7 spectrum. Based on the energy and angular distributions, obtained for the studied H-2(He-8, He-3) H-7 reaction, the weakly populated 2.2(5)-MeV peak is ascribed to the H-7 ground state. It is highly plausible that the firmly ascertained 5.5(3)-MeV state is the 5/2(+) member of the H-7 excitation 5/2(+)-3/2(+) doublet, built on the 2(+) configuration of valence neutrons. The supposed 7.5-MeV state can be another member of this doublet, which could not be resolved in Bezbakh et al. [Phys. Rev. Lett. 124, 022502 (2020)]. Consequently, the two doublet members appeared in the spectrum of H-7 in the work mentioned above as a single broad 6.5-MeV peak.