There are a couple of reasons why a volunteer may not appear in the "assigned to" lists on various pages.
Inactive
If the volunteer's contact record has a System Status of "Inactive", the volunteer will not appear. Update their contact record to make them active again.
Contact Type
Check the volunteer's contact record to see if their "Contact Type" is "Staff", "Clergy", or "Volunteer". If a contact is "Client" or "Associated", they will not appear in the "assigned to" lists.