The primary objective of an oncology dose-escalation trial is to determine the maximum tolerated dose of an investigational drug within a short dose-limiting toxicity (DLT) window. With the growth of immunotherapies in recent years, late-onset toxicity is commonly seen, leading to the elongation of the DLT window. A few time-to-event (TITE) designs have been proposed to discount the information of the patients who have not experienced DLTs and have not completed the DLT window by a somehow arbitrary weight in the binomial likelihood setting. In this work, we propose a new type of Bayesian designs, the DTITE designs, which directly use the time-to-event model to make inference with the calibration-free prior setting. The concept of weight and the binomial likelihood are abandoned, and the time component is incorporated into the DLT probability estimation. Simulation study will be presented to compare the DTITE designs and other TITE designs. The results demonstrate that our DTITE designs can successfully improve dosage targeting efficiency and guard against excess toxicity over a variety of true model settings.