The system will check every 15 minutes until it detects that only 1 card drop remains, then it will start checking every 5 minutes. As a result, Idle Master should move to the next game automatically within 5 minutes of your last card dropping.

You can also press the link that says the number of card drops remaining to manually check.

When I last used Idle Master, once there was only one drop remaining, it automatically switched the check to 5 minutes. Is there a reason you'd need it to drop down to 5 minutes before then?