Hi,
From a Finesse perspective, the SST API that you are referencing will not work when/after the call is terminated. The only workaround you can do is that you can build a custom gadget that has a button that calls the SST API, but the agent needs to be trained to push that button rather than the existing end call button (assuming the existing call control gadget is still visible).
There is a similar sample gadget that is a click to call (Uses the Make call API): https://github.com/CiscoDevNet/finesse-sample-code/tree/master/ClickToCallSampleGadget
You can use the dialog's initiateDirectTransfer JS API: https://pubhub.devnetcloud.com/media/finesse/docs/guides/javascript-library/symbols/finesse.restservices.Dialog.html#initiateDirectTransfer
Thanx,
Denise