Create a 'call me' form on your website that contains a department selector. ThinkAutomation receives the results. It calls back the customer using the number they have provided and speaks 'Thank you for contacting us I am connecting you'.
It then calls the relevant department. If answered it connects the call. If not it sends an email and SMS message back to the customer informing them that the department is currently busy.
It then emails the department asking them to contact the customer as soon as possible.