The constrained Next-to-Minimal Supersymmetric StandardModel (cNMSSM) with mSugra-like boundary conditions at the GUT scale implies a singlino-like LSP with a mass just a few GeV below a stau NLSP. Hence, most of the squark/gluino decay cascades contain two τ leptons. The gluino mass ≳ 1 . 2 TeV is somewhat larger than the squark masses of ≳ 1 TeV. We simulate signal and background events for such a scenario at the LHC, and propose cuts on the transverse momenta of two jets, the missing transverse energy and the transverse momentum of a hadronically decaying τ lepton. This dedicated analysis allows to improve on the results of generic supersymmetry searches for a large part of the parameter space of the cNMSSM. The distribution of the effective mass and the signal rate provide sensitivity to distinguish the cNMSSM from the constrained Minimal Supersymmetric Standard Model in the stau-coannihilation region.