Quantum information science is an emerging field that applies the quantum coherence and correlation to cause the revolutionary advances in computation, communication, and fundamental quantum science. As an irreducible ingredient, Rydberg quantum gate is considered to be a powerful resource with great promises to a wide range of quantum information tasks far beyond the original gate proposals, since the remarkable features characterized by Rydberg atom are long lifetime and giant polarizability. In recent years, the research mainly focused on the properties of Rydberg atom, especially for the case where the effects of Rydberg blockade and antiblockade involving single level for each atom are dominated by van der Waals forces. However, with the variation of interatomic distance, Rydberg interactions can induce more complicated dynamical behavior. This paper studies the implementation of controlled-phase gate and swap gate in one step based on the constructed Rydberg antiblockade (RAB) and double antiblockade (RDAB) regimes when the interatomic distance is less than the characteristic length. Different from the conventional RAB regime that requires weak Rydberg-Rydberg interaction (RRI), our attainable strategy is to compensate the RRI-induced energy shift by properly tuning the detuning between the driving field and atomic transition frequencies. In addition, the proposed RDAB mechanism is a new physical insight that can enable two pairs of Rydberg states belonging to different atoms to be excited, simultaneously. In contrast to other blocking schemes or the schemes without requirements for strong interactions, the merits of our proposal lies in the strong dipole-dipole interaction between two atoms, leading to the population exchange of multiple energy levels. Numerical simulations show that the time evolution of the population for collective double-atom basis obtained from the original Hamiltonian agrees well with the analytical results given by the effective Hamiltonian. In the ideal case, the average fidelity of the controlled-phase (swap) gate can reach 99.35% (99.67%) at final time $t=\sqrt{2}\pi\Delta/\Omega^{2}$ ( $t=2\pi\Delta/\Omega^{2}$ ), and our protocol is robustness against spontaneous emission of high-lying Rydberg states. We believe our present investigation is feasible in upcoming experimental realization and may offer an new venue with respect to on-demand design of new types of effective Rydberg quantum gate devices.