Historically, the primary objective of Phase I clinical trials has been to pick an optimal dose in terms of patient safety, referred to as the maximum tolerated dose (MTD). Most of these trials recommend a “one-size-fits-all” dose for the patient population being studied, while also solely focusing on short-term adverse events. Often patient heterogeneity exists so that group-specific dose selection is of interest. To address the issue of patient heterogeneity, several dose-finding methods have been proposed, including the shift model framework based on the Continual Reassessment Method (CRM). Additionally, for many cancer therapies, relevant toxicities may be defined by participants experiencing adverse events at any point over a long evaluation window, resulting in pending outcomes when new participants need to be assigned a dose. By leveraging the CRM, the time-to-event continual reassessment method (TITE-CRM) provides a feasible approach for addressing this issue. Motivated by a Phase I trial involving radiotherapy that included two patient groups conducted at the University of Virginia, we have developed a hybrid design that combines elements from the TITE-CRM and the shift model framework. This approach helps address patient heterogeneity and late-onset toxicity simultaneously. We illustrate how to perform a dose-finding trial using the proposed design, and compare its operating characteristics to other suggested methods in the field by conducting a simulation study. The shift model TITE-CRM is shown to be a practical design with good operating characteristics in regard to selecting the correct MTD in each group. An R package is also being developed, allowing investigators to provide group-specific MTD recommendations by applying the proposed design, in addition to providing operating characteristics for custom simulation settings.