No, you should not need more space. Sending out an email every day to hundreds of thousands of users should not result in more than 1GB of transfer used. You only need enough disk space to store your users. A large list of 100,000 users should not use more than 100MB space, even with lots of custom data. In fact it could use significantly less space than this.
I see your trouble ticket in the support area but I am at a loss as to what to check. Troubleshooting this directly would require me to make some debug changes and then run dailymail a few times, which I know you don't want to do... A question: Did it work as expected before, then stop working?
Regards