Online Reinforcement Learning (RL) is typically framed as the process of minimizing cumulative regret (CR) through interactions with an unknown environment. However, real-world RL applications usually involve a sequence of tasks, and the data collected in the first task is used to warm-start the second task. The performance of the warm-start policy is measured by simple regret (SR). While minimizing both CR and SR is generally a conflicting objective, previous research has shown that in stationary environments, both can be optimized in terms of the duration of the task, $T$. In practice, however, in real-world applications, human-in-the-loop decisions between tasks often results in non-stationarity. For instance, in clinical trials, scientists may adjust target health outcomes between implementations. Our results show that task non-stationarity leads to a more restrictive trade-off between CR and SR. To balance these competing goals, the algorithm must explore excessively, leading to a CR bound worse than the typical optimal rate of $T^{1/2}$. These findings are practically significant, indicating that increased exploration is necessary in non-stationary environments to accommodate task changes, impacting the design of RL algorithms in fields such as healthcare and beyond.