My wife traveled 3 times this year with the extension letter to Ecuador. The first time was around January, and I was with her. At the airport, they always ask for the letter. They seem to be very updated on this matter. Upon arrival at Miami airport, she showed her expired residence card to the officer, and he asked for the extension letter. No issues at all. Next time in August, she traveled alone. She returned just a few days later. She showed her expired residence card, and the officer called support, and she was escorted to a room. She waited for 20 minutes. Another officer came and asked her if she had the extension letter, to which she replied, "Yes." She was told she had to show the letter, and she replied, but the CBP officer did not ask for it. She was told to always show both documents together to avoid any misunderstandings. She was on her mind like "okkkkkkkkk, whateer you say". Last time, she traveled alone and presented both documents, and everything was just fine. Some officers are just ignorant or plain jerks.
So, in summary, show both documents by default and do not think they will "guide" you for an easy pass.