I use webmail a lot when travelling.
To overcome your problem, I either -
1) manually move my "Sent" mails to the webmail Inbox so they can be collected by Outlook.
2) bcc with your email so they will appear in your inbox and be collected by Outlook.
You can usually only send using the SMTP of the ISP but can collect any POP3 mail.
If you are connected to another ISP for some time, use their email for sending but in setup change the email to your home email. I also do this at home. My ISP is Plusnet so I have to use this for collecting and sending emails, but I have several email addresses that I use from different ISPs and so I just set up each account with the required outgoing email, even though it is using Plusnet's SMTP.
Plusnet Plusnet Pro 21CN Netgear DGND3300v2 router
Windows XP Pro SP3