Communication with your users is vital to ensuring that the help desk works. The more they know about what is happening, the less likely they are to complain. When I was working in support for a major software company, I would call the customer whenever I had any new information and at a very minimum once a week.
Whenever the call needs to be escalated or handed off, the user must be informed. No matterwhat level it gets escalated to, you have to keep the user informed. In general, the flow to the third level is the same as for the second level. Here, too, you need to make sure that the useris informed.
Escalation to the fourth level (that is, external companies) must always have a trouble ticket. One thing that needs to get added to the problem description is the call number from the other company (i.e., the vendor). Without it, it’s difficult to associate their call with yours. You should also ask them to include your call number in their problem description. If they call back and the person who made the call is unavailable, both sides can find the right call.