For an incoming call, my URL gets a GET call which has the recording URL. Is there a way to get the URL of the outgoing call recording as well?

You can use the statuscallback parameter in the outgoing call API and define the statuscallbackevents to terminal which would trigger the call details once the call is completed. Please refer here for more details. An alternate way is to download the call reports (which is a CSV file) and parse them.

You can also implement the call details API for single or bulk calls to access the call details and the recordings. Please refer here for more details.

Note: Reports do not have real-time data. Read more about reports here.