The low-pressure phase diagram of La₃Ni₂O₇ provides an important reference for understanding its pressure-induced high-temperature superconductivity. While the spin-density-wave transition at T_SDW≈150 K is increasingly well established, the origin of the second density-wave transition at T_DW≈130 K has remained unresolved. Here, we perform unrestricted Hartree-Fock calculations to investigate the potential origin of the second transition. {Within the orthorhombic phase, the degeneracy between possible ordering wavevectors at Q_Y=(0,π) and at Q_X=(π,0) is lifted and the electronic system} develops a double-stripe spin-density wave with ordering vector Q_Y=(0,π). We identify that the pure double stripe spin state is unstable in La₃Ni₂O₇ towards a commensurate charge-density wave instability, which favors a spin-modulated double stripe order with intertwined charge and spin instabilities and establish the hierarchy of ordered states in La₃Ni₂O₇. We further discuss our results in the context of available experimental literature and propose further experimental tests to elucidate the origin of the SDW/DW states in this system.