This issue could be localized to just our instance but I have noticed the “New Documents” do not update with the new associated content types in Document Sets. Typically what happens is after adding a (custom) Content Type into a Document Set. You have to click “Update the Document Set” in order for the new Content Type to show up. What is most odd is at least one Document Set behaves this way and the rest do not.
After numerous re-instancing, I found out the Document Set must be marked approved for the rest of the Document Sets to update. I still cannot explain why there was one Document Set that didn’t have to meet this condition. I suspect that the Document or Content Type properties are different leading to this inconsistent behaviour. Turning Content Approval off avoids this issue. That isn’t a viable solution for me.
This is the text you would see within each of the Document Sets after recently adding or subtracting a content type:
Content types that are available to this Document Set have been added or removed. Update the Document Set.