Predictive process monitoring (PPM) offers multiple benefits for enterprises, e.g., the early planning of resources. Its efficacy depends on the quality of event data used for model training. In this work, we study the effects of unseen behavior, i.e., events that are not present in the training data, on prediction quality. Unseen behavior might occur due to infrequent traces or added compliance constraints. Existing approaches focus on predicting unseen behavior based on updating the prediction model. Another option is to inject unseen behavior into the training data based on order and temporal constraints on events. Due to the model-agnostic nature of log injection, different PPM approaches can be employed without any modification. The proposed algorithms are prototypically implemented and evaluated on real-life event logs. The results demonstrate that log injection can enhance prediction quality and is more time-efficient than state-of-the-art model update strategies.