Consider defining default Sharing Rules within the Access Level. In the example below, the default has the user with this access sharing with their home group and granting them view permission. As a system admin, you may want to change this to something else or add more rules to meet your organizations sharing needs.
Also Project Managers can define project access template , which will be used for the future projects. Who ever was on the sharing list (see access levels above) will always get access to the new projects after PM has hit the Set as my Project Access Template.