What options do I have to extend my stay in Thailand after starting a 30-day extension with a letter from the UK embassy?

IF you already used your "regular 30 day extension" (which you could have gotten from a 30 day visa exempt or a 60 day tourist visa entry) your only option is the 'unable to travel due to covid-19' extension.

You sound a LOT like you haven't gone to get that extension yet <- seeing as NO ONE gets 30 days out of the gate when they apply for it :O Most offices are given a 15 day "under consideration" stamp and telling you to come back when that date is up to get the other 15 days that the "up to 30 day" extension based on embassy request can give you.

DID you apply for the covid-19 extension yet?

You can probably apply for a second letter at the end of the month from the embassy.

As you waited until the last possible moment, the time to take action and possible conversion to other visa is gone. Sorry, no more options left than leave, wait out this Covid situation and apply for a long term visa for your next visit