We report a measurement of the CP-violation parameter sin2beta with B^0 -> J/psi K^0_S decays in which the J/psi decays to hadrons or to muons that do not satisfy our standard identification criteria. With a sample of 88 million BBbar events collected by the BaBar detector at the PEP-II asymmetric-energy $B$ factory at SLAC, we reconstruct 100 +/- 17 such events, with J/psi -> pi^+ pi^- pi^0 being the most prevalent, and measure sin2beta = 1.56 +/- 0.42 (stat.) +/- 0.21 (syst.).