This is only semi-possible right now.
Studio Flow executions are a bit different from normal webhooks that return TwiML, since they MUST be the first webhook hit by the call. (For voice calls, it is also a best practice to make sure the same Studio Webhook is configured as the secondary handler and call status handler for the phone number in question, which suggests that Studio is a bit more robust in how it operates than a typical TwiML App.)
The workaround is to use the <Dial>
verb to create a new call to the Studio Flow that you wish to redirect to. This assumes that each Flow that handles incoming voice has its own phone number(s).
That said, I have been told that a feature request is underway for sometime in 2021 that would allow for different flows to be linked together, so this may change in the future.
I am not exactly sure how you specify a webhook for once the Flex agent is finished with the call -- it looks like you have already done some research into the TaskRouter functionality that enables this. But assuming you can specify a webhook URL (assignment_redirect
in your example), all you should need to do is create a TwiML Bin containing your <Dial>
verb to the satisfaction survey Studio flow you want to receive the call, and use that TwiML bin as the webhook URL.