You can do this in Task Phases. Go to the Model/Tasks tab, and under automaticlly open tasks fill in the category of the task you want to open and in the condition you can use status in $file="closed". You can also set the background (true or false) to either pop-ul the task or just open it in the bg.
There is not an OOB method to trigger a new task based on the previous task closing prior to Process Designer change management (which requires 9.3x SM versions).
The tradtional method would have been to design a task category with multiple phases, so that the first group completed their phase, then at phase closure, the second group could work on the task. For example, a task category could have the phases: Provision VM Licsense -> Configure VM -> Validate with Customer
You could potentially configure a wizard that would walk the current logged in user through opening a new task when the task was closed (based on some conditional evaluation of the current task at closure). If needed, it could execute in the background.