IVR transfer to call center

I am working on an IVR application that has some API calls to a customer care database.

I suspect that there is some way that if I have figured out the customer, and they transfer to an customer service rep, for me to pass information about who the customer is so the CSR does not have to ask them for it again. How is this usually done? Is there a name for this feature so I can google some options? I was thinking I could stick a customer id in some extra SIP parameter, and then customize the CSR’s soft phone so that it would recognize the parameter, and maybe even build a dynamic URL that would open the relevant record in the customer care database.