Using a data sample of 106 million psi(3686) events collected with the BESIII detector operated at the BEPCII storage ring, we study for the first time the decays chi(cJ) -> phi(KSK +/-)-K-0 pi(-/+) and chi(cJ) -> phi K+K-pi(0) in the E1 radiative transition psi(3686) -> gamma chi(cJ). The decays are dominated by the three-body decay chi(cJ) -> phi K*(892)(K) over bar. We measure branching fractions for this reaction via the neutral and charged K*(892) and find them consistent with each other within the expectation of isospin symmetry. In the K (K) over bar pi invariant mass distribution, a structure near the K*(892)(K) over bar mass threshold is observed, and the corresponding mass and width are measured to be 1412 +/- 4(stat) +/- 8(sys) MeV/c(2) and Gamma = 84 +/- 12(stat) +/- 40(sys) MeV, respectively. The observed state favors an assignment to the h(1)(1380), considering its possible J(PC) and comparing its mass, width and decay mode to those reported in the Particle Data Group.