Allow the pop out button on all screens of the extension and not just the main. Or for all card related searches persist the card screen until manually closed.
The pop out button is available on all screens of the extension in the beta preview; please let me know if you are not experiencing this.
In addition, we are planning to persist the last visited page when the popout closes (not currently available in the beta preview but coming soon). This should make the workflow you mentioned even easier as it will keep the Card open even if you forget to use the pop out button.