In the open online dial-a-ride problem, a single server has to deliver transportation requests appearing over time in some metric space, subject to minimizing the completion time. We improve on the best known upper bounds on the competitive ratio on general metric spaces and on the half-line, for both the preemptive and non-preemptive version of the problem. We achieve this by revisiting the algorithm $\textsc{Lazy}$ recently suggested in [WAOA, 2022] and giving an improved and tight analysis. More precisely, we show that it has competitive ratio $2.457$ on general metric spaces and $2.366$ on the half-line. This is the first upper bound that beats known lower bounds of 2.5 for schedule-based algorithms as well as the natural $\textsc{Replan}$ algorithm.