We report the measurement of the inclusive cross sections for e⁺e^- → nOCH (where nOCH denotes non-open charm hadrons) with improved precision at center-of-mass (c.m.) energies from 3.645 to 3.871 GeV. We observe three resonances: R(3760), R(3780), and R(3810) with significances of 8.1σ, 13.7σ, and 8.8σ, respectively. The R(3810) state is observed for the first time, while the R(3760) and R(3780) states are observed for the first time in the nOCH cross sections. Two sets of resonance parameters describe the energy-dependent line shape of the cross sections well. In set I [set II], the R(3810) state has mass (3805.7 ± 1.1 ± 2.7) [(3805.7 ± 1.1 ± 2.7)] MeV/c², total width (11.6 ± 2.9 ± 1.9) =[(11.5 ± 2.8 ± 1.9)] MeV, and an electronic width multiplied by the nOCH decay branching fraction of (10.9 ± 3.8 ± 2.5) [(11.0 ± 3.4 ± 2.5)] eV. In addition, we measure the branching fractions B=R(3760) → nOCH] = (25.2 ± 16.1 ± 30.4)% [(6.4 ± 4.8 ± 7.7)%] and B[R(3780) → nOCH] = (12.3 ± 6.6 ± 8.3)% [10.4 ± 4.8 ± 7.0)%] for the first time. The R(3760) state can be interpreted as an open-charm (OC) molecular state, but containing a simple four-quark state component. The R(3810) state can be interpreted as a hadrocharmonium state.