Question:

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?

Answer:
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