transfer the call to a remote agent instead of just an extension. remote agents do not have computers (don't let the name fool you, remote agents in vicidial have nothing to do with location, it's just "no computer").
Advantages:
The user assigned to the remote agent will then be visible in real time on that call until the call completes.
You'll get reporting and recording.
Transfer can be done with Ingroup and not extensions. Each ingroup has a name instead of a 10 digit number. You CAN create an ingroup per Remote Agent if you require precise choice during transfer.
If a number for one of the remote agents changes, the live agents never need to know or care. Just change it in the Remote Agent interface and you're done.
Remote agents have an interface. If you require the ability for them to turn themselves on/off, that can be done.
If you have multiple users at one extension, you can have multiple lines for the remote agent. (You lose "individuality" that way, but if you create enough sequential users at least you'll see that the calls are in progress and your reporting will be valid in the aggregate, just individual user reports will be meaningless without a method to identify each agent by extension.)
alternately
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