Try this first: Go into Accounts, click More Settings, then select the Outgoing Server tab. Check "My outgoing server requires authentication / Use same settings as my incoming email server". Keep in mind that this might not work for everyone. The ability to send outgoing emails from a different domain varies from ISP to ISP.
As a workaround, you can set up a new split account for this situation. Create a new email account named something like "ROAMING SEND". You want to copy over all the details from your normal account, but change Outgoing mail server (SMTP) to "smtp.gmail.com". In More Settings, check "My outgoing server requires authentication / Log on using" and enter your Gmail credentials. The advanced tab SMTP should be set to port 587/TLS.
Next login to Gmail and click Settings at the top right of the window. Select the Accounts tab and in the Send Mail As section you want to enter your email address. Make this the default, and specify Send through Gmail, and Always reply from the default address.
When you compose new Outlook messages while roaming outside your domain, just select the ROAMING SEND account in the dropdown menu at the top of the email. Recipients of your emails will see that your message is coming from your normal account, but relayed via Gmail. When they reply to the message, it will be directed to your normal address (not Gmail).
Hope this helps!