TODO:
When you click on the Team Site link in Office 365, it takes you to https://sitename.com/SitePages/Home.aspx. The issue is, that we really want to go to https://sitename.com/site/Intranet/SitePages/Home.aspx.e.aspx.
SOLUTION:
To accomplish this, I added a CEWP to the sitename.com/SitePages/Home.aspx, that has a simple script that redirects to the page I really want to go to.
<script teyp="text/javascript">window.location.href = "https://sitename.com/sites/Intranet/SitePages/Home.aspx";</script>
NOTES:
So now the obvious, how do I ever edit the page again? To do this I used SharePoint Designer. Simply open the site by placing the root url in the location to open input (sitename/SitePages/). From there I can now edit the page and the CEWP.