We have constructed numerically non-Abelian vortices in an SU(2) Chern-Simons-Higgs theory with a quartic Higgs potential.We have analyzed these solutions in detail by means of improved numerical codes and found some unexpected features we did not find when a sixth-order Higgs potential was used. The generic non-Abelian solutions have been generated by using their corresponding Abelian counterparts as initial guess. Typically, the energy of the non-Abelian solutions is lower than that of the corresponding Abelian one (except in certain regions of the parameter space). Regarding the angular momentum, the Abelian solutions possess the maximal value, although there exist non-Abelian solutions which reach that maximal value too. In order to classify the solutions it is useful to consider the non-Abelian solutions with asymptotically vanishing At component of the gauge potential, which may be labeled by an integer number m. For vortex number n = 3 and above, we have found uniqueness violation: two different non-Abelian solutions with all the global charges equal. Finally, we have investigated the limit of infinite Higgs self-coupling parameter and found a piecewise Regge-like relation between the energy and the angular momentum.