Yes You can take Exit permit by Jun 30 and it is valid for 7 days.
Also if the Computer card is expired, even then, you can take exit permit.
Your Mandoub will have to pay some amount like 250 Qr something and the computer card will become active for that day only. Then he can apply for exit permit for you and for other employees and the next day, the computer card will expire again. Most companies will not reveal this info to employees.
Hope this info helps you.