In Outlook, choose Tools, Account Settings. Double-click your email account. Change only the Outgoing mail server (SMTP) to what your temporary Internet connection uses. If the mail server allows anyone to send from within the network, you're finished. Click Test Account Settings to see if Outlook can send mail.

In my case, Earthlink required authentication. If needed, click More Settings, and click the Outgoing Server tab. Activate the checkbox at the top, and click the radio button to Log on using. Enter the ISP's mail username and password. Click OK, and click Test Settings to see if everything works.

Now mail will be sent using the local ISP's mail server. Recipients can still see detailed Internet headers and identify that local ISP, but without manually investigating, the email will look like it came from your regular account. They'll just reply as they normally would. Just reset the settings later when you return home.