How can I extend my Covid visa while waiting for a Non-B visa application in Thailand?

Unfortunately there is no short term extension :/ when you can't get the documentation in time.

That leaves you with two options,

Apply for another 60 day covid extension

OR

Apply for a 60 day visit thai family extension <- which you can do ONCE per entry (so you might wanna save this one and go for the covid extension instead)

As

@Bra****

said, just go apply for another 60 day covid extension.

Now whether it's "painless" to go from an ED visa extension (which hopefully you cancelled before you switched to a covid extension) to an in country Non-B 90 day visa is totally gonna hinge on whether your employer can meet the document requirements at the immigration office you're applying at AND if you have enough time on your current stamp to get thru the under consideration period.

SO you need to be talking to your future employer and the immigration office you'll be dealing with for those answers.

You don't need to do anything special. Just go and apply for another Covid extension and you'll get 60 more days.