I have today been in contact with Telstra today about this issue. After an online chat that went for about 45 minutes and that involved 'chatting' with 3 different people (as I was transferred from area to area), I finally got to 'chat' to someone who told me this is definitely a limitation of the old type of mail account. (It is amazing how little the consultants understood the problem that is faced by so many people who have had Bigpond email address since before 2012.)
The advice from one consultant was that, if you have this old style of account, you need to send your mail using a different SMTP server when you are not using a Telstra connection - which is something that the majority of people would not have a clue how to do, and is not really feasible when you are travelling.
Instead, the only way to effectively solve this problem for good is to update your 'old-style' Bigpond email account with Telstra to the 'new-style' email account that they now offer, which is hosted by Microsoft Outlook. This new style of account will allow you to send your mail from anywhere, as long as you have an internet connection via WiFi or Cellular.
Here is the link to the web page that gives you the option to perform this Bigpond account upgrade.
You will need to make sure you initiate the upgrade at a time when it is not critical for you to have access to your email, as it can take up to 24 hours for the upgrade to complete.
But just be warned that, after the upgrade, you will need to change your email setup on any computer, and install the new mail account on your tablet and mobile devices.
If it is important to you that you keep your old Bigpond account Inbox and Sent mail, make sure you just deactivate that account and install the upgraded mail account as a new email account.
Videos from Telstra are included below, showing you how to install your new type of Bigpond mail account on your computers and tablet/mobile devices.
It is well worth considering doing this upgrade if you are going to be travelling and want to be able to use your Bigpond email account to sent emails while away.
If you need help with this upgrade, or need further advice on this topic (or if the above just sounds like 'gobbledy-gook'), please don't hesitate to contact iTandCoffee on 1300 885 420 or email@example.com.