Why don't you link the contact to a company? Each ticket has a contact, and this contact works for a company. That link between contact and company can be maintained by editing the contact.  

We are a small company, and the reporter may also be an agent in Freshdesk.  It appears you can't have a company assigned to a Freshdesk agent.

Are you saying that you don't inform anyone else (apart from the agent/reporter) when an issue is fixed?

Our agents also raise tickets, but these are recorded with the email address of a customer as contact, and then get the type "Phone". Then the contact can be linked to a company.