Indeed. I don't believe it's possible with standard installation to do this for all services.
Probably not what you're looking for, however Exim as a service is configurable to achieve this via /etc/mailips on a manual per-domain basis. More information on that can be found in the documentation here.
Are you attempting to modify the IP address used for all outgoing connections, or just when sending email? There is an option in the Basic Exim Configuration Editor within WHM to send mail from IP addresses delegated to accounts with dedicated IPs.