So two bits I think.

1) Get SSPR to display the data.

2) Permissions to modify

I am pretty sure SSPR can do more attributes it is in the Helpdesk module settings. Then make sure that either the account your helpdesk team login with, or the proxy user have sufficient eDir permissions to W to the attributes.

1) Get SSPR to display the data.

It is done. The Helpdesk can see the phone attribute on the found user's profile tab.

2) Permissions to modify

I understand to need to add permission to the attribute in eDirectory (for the Helpdesk or the proxy technical user). But where can I configure that the Helpdesk can modify this from SSPR Helpdesk module?

I don't think you can do this from SSPR.

If you have Advanced Edition of IDM you can use UA Dashboard for this or write a custom Workflow form for it depending on what you need.

Mostly I have created a simple (or advanced) form for helpdesk members to edit users.

I don;t have an SSPR I am admin on to look at.  I thought there was a user update module in SSPR, or am I remembering wrong? 

Yes, there is an 'Update Profile' module in SSPR. The user can modify/update his own profile data.

In Helpdesk module only can change password and isn't possible to add attributes what can Helpdesk user can modify.