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?
There is no direct programmatic way of getting hold of an outgoing call recording URL. One easy way is to download the call reports (which is a CSV file) and parse it.
PS: Reports do not have real-time data. Data is available only for the previous working day. Read more about reports here