The instance does not have an ipv4 IP address. The IP in question is from the underlying host your VM runs on. So there’s no predictable way to get a new outbound IP except moving regions, for now. We’d love to offer fixed outbound IPs in the future.
For now, you could use a proxy. MyProxyservice has cheap plans with fixed IPs. Or, you could run a simple proxy inside Fly, which could be moved around more easily if one of our IPs causes trouble.