@Jonathan *******
There's nothing that says you can't use that one time 60 day visit thai family extension to "buy time" to apply for another extension.
I see it used routinely when people don't meet the seasoning requirements for retirement/marriage to a thai extensions, they get that 60 days let their funds season and then apply for the year without any issue
It SHOULD work that same way for you, cancel your extension based on employment, apply for the 60 day visit thai family extension, get the documentation from the new employer and then apply for the new extension based on employment.
Sincerely good luck with it (y)