Summary:
If you want to give a regular timesheet user access to a Specific project where he is not the project leader, then the only way is to assign this user as a Substitute User to the project leader.

Note:
Enabling a user as a substitute user (to Project Leader) will provide user the visibility to all the projects belonging to that Project Leader. When you assign the user as a substitute user to the project leader, ensure that you grant him Full Access, else he will not have access to the Projects of the Project leader. Forward Notifications is optional.