We study the tunneling ionization (TI) of Ni, Ni+, and Ni2+ with a time-dependent density-functional-theory method and reproduce the puzzling suppression of the TI of Ni and Ni+ and the enhancement of TI in Ni2+. Numerical results reveal that for all three species the electron tunnels from a 4s orbital; that is, excitation precedes tunneling for both of the cations, for which the highest orbitals are 3d. The effective radial potentials for the d orbitals have a centrifugal barrier, while there is no such barrier for the s orbitals. At the classical turning point for the 3d orbital, the 3d to 4s excitation energy is lower than the centrifugal potential for the d orbitals. Two factors of opposite nature are identified in this work. On the one hand, electrons moving away from the nucleus in the intense laser fields induce an attractive potential that effectively lowers the energy level and thus suppresses tunneling. Excitation, on the other hand, has the opposite effect and enhances tunneling. The energy gap between 4s and 3d is small for Ni+ and therefore suppression wins. As the charge of the cation increases, the excitation energy becomes much greater, and for Ni2+ enhancement dominates. Based on a similar analysis, we expect enhanced TI for several transition-metal cations of charge 2 and higher.