We revisit the well-known online traveling salesman problem (OLTSP) and its extension, the online dial-a-ride problem (OLDARP). A server starting at a designated origin in a metric space, is required to serve online requests, and return to the origin such that the completion time is minimized. The SmartStart algorithm, introduced by Ascheuer et al., incorporates a waiting approach into an online schedule-based algorithm and attains the optimal upper bound of 2 for the OLTSP and the OLDARP if each schedule is optimal. Using the Christofides' heuristic to approximate each schedule leads to the currently best upper bound of (7 + sqrt(13)) / 4 approximately 2.6514 in polynomial time. In this study, we investigate how an online algorithm with predictions, a recent popular framework (i.e. the so-called learning-augmented algorithms), can be used to improve the best competitive ratio in polynomial time. In particular, we develop a waiting strategy with online predictions, each of which is only a binary decision-making for every schedule in a whole route, rather than forecasting an entire set of requests in the beginning (i.e. offline predictions). That is, it does not require knowing the number of requests in advance. The proposed online schedule-based algorithm can achieve 1.1514 * lambda + 1.5-consistency and 1.5 + 1.5 / (2.3028 * lambda - 1)-robustness in polynomial time, where lambda lies in the interval (1/theta, 1] and theta is set to (1 + sqrt(13)) / 2 approximately 2.3028. The best consistency tends to approach to 2 when lambda is close to 1/theta. Meanwhile, we show any online schedule-based algorithms cannot derive a competitive ratio of less than 2 even with perfect online predictions.