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If it's not working for you, you may need to get the Outgoing settings for the ISP in the hotel, as it may only allow you to send using their outgoing server.
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This is a common (and very frustrating) problem for people who travel a lot ,particularly overseas.
The main problem is that your email address is possibly on an older legacy "Client/Server" type email system.
While you can manage to get arround it by using Telstra/Big Pond "web mail", I find this difficult on an iPhone and rather "clunky" to work with.
As you have an iPhone, you most likely have an iCloud account (or another web based email account such as gmail, or hotmail?)
My simplest solution is to set a web based mail server such as iCloud as your outgoing mail server by: (Note - THE FOLLOWING UPPER CASE IS THE IPHONE SCREEN DISPLAY OPTION, WHICH MAY NOT BE IN CAPITALS ON THE SCREEN
MAIL, CONTACTS, CALENDARS >
ACCOUNTS - select one, preferably the iCloud one, but if it is not there ADD ACCOUNT > and follow prompts
OUTGOING MAIL SERVER
SMPT (ICLOUD or other) >
At this point you should be looking at your PRIMARY SERVER, and OTHER SMPT SERVERS, turn the mail.bigpond.com OFF >, then turn the iCloud (or other) ON >
The downside of this is that emails from your iPhone will come from your iCloud address, but it does mean that you can send emails from Wi-Fi locations without any issues. To keep a copy of such emails always BCC your bigpond email (there is a setting to do this automatically)
As a reasonably frequent overseas traveller I have given up on Wi-Fi and always get a Casual International data pack, they work particularly well, they are good value, and you avoid all the queues at the free wifi cafes etc
I hope this has been of some help
ps - 587 was the right number!!