  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.
