How to improve support service in any organization?

You need to implement a "full life-cycle for service support" based on a "service strategy" that serves as a foundation for the entire life-cycle.    You also need a good "governance" and a good "measurement/reporting" system to complement the service model.  Depending on the specifics of your situation, you may also need a service catalog for your support services, and the agreed service level agreement and its service level targets per each listed service, and the entitlements to different customers based on the their classification and categorization within the priority list 

All of the above is needed in the front-end. However, the more challenging part may be in the back-end where you have your dependencies on other functions/departments throughout the organization (internal dependencies) and on external entities/suppliers/ ... (external entities). May I suggest you go and study ITIL documentation to grasp the model  and focus on the service support part as it seems to be more directly related to what you are asking about !