Summary: Most practical scheduling applications involve some uncertainty about the arriving times and lengths of the jobs. Stochastic online scheduling is a well-established model capturing this. Here the arrivals occur online, while the processing times are random. For this model, Gupta, Moseley, Uetz, and Xie recently devised an efficient policy for non-preemptive scheduling on unrelated machines with the objective to minimize the expected total weighted completion time. We improve upon this policy by adroitly combining greedy job assignment with \(\alpha_j\)-point scheduling on each machine. In this way we obtain a \((3+\sqrt{5})(2+\Delta)\)-competitive deterministic and an \((8+4\Delta)\)-competitive randomized stochastic online scheduling policy, where \(\Delta\) is an upper bound on the squared coefficients of variation of the processing times. We also give constant performance guarantees for these policies within the class of all fixed-assignment policies. The \(\alpha_j\)-point scheduling on a single machine can be enhanced when the upper bound \(\Delta\) is known a priori or the processing times are known to be \(\delta\)-NBUE for some \(\delta\geq 1\). This implies improved competitive ratios for unrelated machines but may also be of independent interest.