Unique IP and outgoing requests

I have a unique IP for one of my domains ( for the sake of discussion). So, pointing a web browser to brings up my website, and similarly I can SSH into that address and access my shell account. In other words, incoming requests work as expected.

However, if I SSH into and look up my IP address, it’s really something like

This is a problem because I would like for outgoing requests from a PHP script on the server to originate from, not Is this possible? This is important because I am trying to use a webservice that has whitelisted. (As a footnote, I’m aware that my unique IP address is not static and may change. This doesn’t matter, as I’m just running some tests, not setting up anything permanent.)