We are not aware of any requirement in the Truth in Savings Act or otherwise mandating that a customer sign a CD. However, we do recommend requiring the customer to sign a new account agreement when switching to a new CD account that has a different term. If your account agreement is included on your CD, then your customer should sign the new CD.
The account agreement is the instrument by which customers bind themselves to the terms of a specific account. When a customer requests a new CD that will have a different term, the customer should sign a new agreement reflecting that new term. In our view, providing the new CD without obtaining a signature risks insufficiently binding the customer to all of the terms of that new account.
We are aware that some institutions have customers sign master CD agreements that cover all CDs purchased by that customer. However, from what you have told us, your institution is not using a master CD agreement, and consequently we recommend obtaining new signatures for each new CD.