The solution was to add a text form field to the first page of the letter that authors use to enter the letter's send date and then to place a cross-reference to the text field in the header of subsequent pages. While authors usually remember to change the date on the letter's first page, they almost always forget to make the same change to the date in the header of following pages. That's why we can't add a field to the letters that displays the current date. In addition, the letters are sometimes sent on dates other than the official 'sent' date.
Unfortunately, the letters are usually composed several days before they're sent, and the send date often shifts.
One of the most-used templates in our collection is a multipage letter that shows the date on the top of the first page, just under the company logo, as well as in the header of subsequent pages. The templates allow us to ensure all the documents we send to clients and partners share a similar look and feel. The company I work for relies on Microsoft Word templates to create all sorts of documents, from one-page letters to reports several hundred pages long.