The new related contacts field in the upper right of a case is great. However, you should be able to click on any of the listed providers and open the contact side-widow to view full contact details, revealing the standard functional contact links.
The quick view should also show email address. Phone and email should be made clickable to take action right from this page.
Currently all you can do is view the provider, but to take futher action, you need to leave, making this function of limited value.