When selecting "Add Dependent" from a Household, to create a NEW contact you are not given the New Contact screen. A contact is simply created without the option to edit the contact prior to creation. If you use permissions to divide your database, this results in a user who is unable to see or edit the new contact they just created.
The two solutions I can think of; Give the option of creating the new dependent via the new contact screen, where the user can set their team permission. OR have the option in settings, to give new dependents created this way, to copy the permissions from the Head of household.