Currently, there it difficult to discover that the quick pick menu is able to be dragged & repositioned, offering no visual indicator to communicate the functionality:

In addition, when the feature has been discovered, it is difficult to know what part of the UI is "grabbable". After some trail & error I discovered that the area in question is the 6px padding around the text input, which is hard to easily grab:

A potential solution to this issue would be to mimic the macOS Spotlight Search behaviour, making the entire text input draggable when empty: