The design of tailored hardware has proven a successful strategy to reduce the timing analysis overhead for (hard) real-time systems. The stack cache is an example of such a design that has been proven to provide good average-case performance, while being easy to analyze. So far, however, the analysis of the stack cache was limited to individual tasks, ignoring aspects related to multitasking. A major drawback of the original stack cache design is that, due to its simplicity, it cannot hold the data of multiple tasks at the same time. Consequently, the entire cache content needs to be saved and restored when a task is preempted. We propose (a) an analysis exploiting the simplicity of the stack cache to bound the overhead induced by task preemption and (b) an extension of the design that allows to (partially) hide the overhead by virtualizing stack caches.