The situation for US citizens as I understand it:
1. Apply for the embassy letter (they basically want data from your passport photo ID page) here:
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2. You should receive the email letter in hours of applying (like the one above with personalized with your data). Print it out, if you can, in color on good paper (not required since Thai Immigration has agreed to accept emailed embassy letters from the US, but can't hurt). You also may want to print the email you received from the US Embassy Bangkok to which the embassy letter was attached. This shows that the US Embassy Bangkok emailed it to you (again shouldn't be required, but ....). Nice to have the response letter from the US Embassy Bangkok on your smartphone and the embassy letter itself (Again, not required, but can't hurt).
3. Apply at your local Thai Immigration office for the 30-day "emergency due to Covid-19" extension. Should cost 1,900 baht (but some places have been asking for 2,000). You will likely only receive an "under consideration stamp" that names a date to return to the immigration office.
4. Return to the immigration office on your return date named in your "under consideration" stamp. Hopefully, you will receive another stamp indicating you have final approval of an extension of stay from 27 September to 26 October. Your continued presence in the country is made legal by the current amnesty to 26 September and/or the return to immigration date stamped in your passport on your "under consideration" stamp.
5. There have been statements from some officials (in the Phuket office) that this will be a repeatable process. That is, nearing the end of your 27 September to 26 October extension acquire another embassy letter from US Embassy Bangkok... basically repeat steps 1-4. There is of course no guarantee that the US Embassy Bangkok will continue to provide the letters and no guarantee that Thai Immigration will continue granting extensions, but that looks like a possibility going forward. No one knows how long that will last.