I don't see the "New" button when trying to add a new picklist value to the Status field on the Campaign Member. The two picklist values that exist now are Responded and Sent. This field is required per the Page Layout.
I have full system admin rights and looked at other standard picklist fieldsand they all have a "new" button available. What am I missing?
To add or change member status values, view the campaign and then select the Advanced Setup button up top next to Edit. Here you can change and add values as well as set the default and Responded attribute.
This was done this way since you can have many campaigns with different status and default values. A single picklist would quickly become too large to work with.
For more information, please see this short blog about Salesforce campaigns.
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