We have over 100 deployments of Live Agent. One for each of our company brands and locations. The auto greeting for each is specific for the brand & location.
Example: Hello! Thanks for chatting with us at My Business of Chicago. My name is {!User_FirstName}.How can we help you today?

When a customer starts the chat with a location then the agent determines they are not who the customer needs to be speaking with and transfers the chat to the correct agent, the original message repeats with the same values as before, even though the agent and location/business unit no longer apply. This leads to customer confusion.

How can we have a greeting and ensure the greeting is correct even through transfers? I'm open to a configuration or coding solution.

Steps to reproduce:
Start a live agent chat session from one of our web pages, say RV Sales of Atlanta
You receive greeting like: Hello! Thanks for chatting with us at My Business of Atlanta My name is John.How can we help you today?
Ask the agent a question about My Business Web Page (buying retail product)
The RV Sales agent will indicate they must transfer you to a different agent (call center)
Chat is transferred, you will receive same greeting: Hello! Thanks for chatting with us at My Business of Atlanta My name is John.How can we help you today?
but should have got a greeting like Hello! Thanks for chatting with us at My Business. My name is Jill. How can we help you today?

During the transfer the Agent Name and Location will both be changed, but the message is the same.

I saw and voted for an Idea about this, but wanted to know how others have addressed this issue?

Thanks.
John.