Using a data sample of 980 fb-1 collected with the Belle detector operating at the KEKB asymmetric-energy e+e- collider, we present evidence for the ω(2012)- in the resonant substructure of ωc0→π+(K̄Ξ)- ((K̄Ξ)-=K-Ξ0+K̄0Ξ-) decays. The significance of the ω(2012)- signal is 4.2σ after considering the systematic uncertainties. The ratio of the branching fraction of ωc0→π+ω(2012)-→π+(K̄Ξ)- relative to that of ωc0→π+ω- is calculated to be 0.220±0.059(stat.)±0.035(syst.). The individual ratios of the branching fractions of the two isospin modes are also determined and found to be B(ωc0→π+ω(2012)-)×B(ω(2012)-→K-Ξ0)/B(ωc0→π+K-Ξ0)=(9.6±3.2(stat.) ±1.8(syst.))% and B(ωc0→π+ω(2012)-)×B(ω(2012)-→K̄0Ξ-)/B(ωc0→π+K̄0Ξ-)=(5.5±2.8(stat.) ±0.7(syst.))%.