Transfer Advanced Routing Chat but need new skill

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We have a bunch of skills routing into a single TechSupport queue.  Agents get assigned chats based on their assigned skills.  Works great!

However, if a chat comes in with product X associated with it and our agent realizes the customer selected the wrong product, we need to transfer that chat to someone else.  If I select a different product Y in the chat workspace and then transfer back to the queue, it ends up coming right back because it didn't change the product/skill and the system thinks it is still a product X chat.

How do I change the skill associated with a chat so that transfers to queue get re-routed based on the newly selected skill?