Hi Chris,
Thanks for your reply.
To let domain-joined users access Office 365 web services without typing any password, you can refer to the following information:
1. For the Office 365 portal:
https://community.office365.com/en-us/wikis/sso/using-smart-links-or-idp-initiated-authentication-with-office-365.aspx
2. For the team site main page:
https://community.office365.com/en-us/forums/613/p/163262/467794.aspx
3. For Outlook Web App (OWA):
You can let end users access http://www.outlook.com/domain.com (domain.com is the current federated domain).
As a tip, to achieve the goal for the Office 365 portal and the team site, we need additional URLs internally, such as http://365portal.domain.com/ and http://365teamsite.domain.com/.
Moreover, I noticed that you have posted another thread about map drives for SharePoint Online and our support engineer is following the issue. If you need further assistance about the map drive issue, please feel free to post it in that thread. Thanks for understanding.
Thanks,
Claud