Shubnikov-de Haas measurements of high quality URu₂Si ₂ single crystals reveal two previously unobserved Fermi surface branches in the so-called hidden order phase. Therefore, about 55% of the enhanced mass is now detected. Under pressure in the antiferromagnetic state, the Shubnikov-de Haas frequencies for magnetic fields applied along the crystalline c axis show little change compared with the zero pressure data. This implies a similar Fermi surface in both the hidden order and antiferromagnetic states, which strongly suggests that the lattice doubling in the antiferromagnetic phase due to the ordering vector Q_AF=(001) already occurs in the hidden order. These measurements provide a good test for existing or future theories of the hidden order parameter.