Should you update the design when you rebrand?
Yes, when the mark changes: menu, poster and profile photo have to show the new one the same day. A cloth that does not carry the name does not need remaking if guests do not use it to recognise you.
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List every piece where the old name can still be read. The sign is new and the laminated menu still carries the logo that no longer exists. You get a list of pieces that change and those that stay, with the new PDF only for the first group.