We perform a search for new physics using final states consisting of three leptons and a large imbalance in transverse momentum resulting from proton-antiproton collisions at 1.96 TeV center-of-mass energy. We use data corresponding to 5.8 fb-1 of integrated luminosity recorded by the CDF II detector at the Tevatron collider. Our main objective is to investigate possible new low-momentum (down to 5 GeV/c) multileptonic final states not investigated by LHC experiments. Relative to previous CDF analyses, we expand the geometric and kinematic coverage of electrons and muons and utilize tau leptons that decay hadronically. Inclusion of tau leptons is particularly important for supersymmetry (SUSY) searches. The results are consistent with standard-model predictions within 1.85σ. By optimizing our event selection to increase sensitivity to the minimal supergravity (mSUGRA) SUSY model, we set limits on the associated production of chargino and next-to-lightest neutralino, the SUSY partners of the electroweak gauge bosons. We exclude cross sections up to 0.1 pb and chargino masses up to 168 GeV/c2 at 95% C.L., for a suitable set of mSUGRA parameters. We also exclude a region of the two-dimensional space of the masses of the neutralino and the supersymmetric partner of the tau lepton, not previously excluded at the Tevatron.