I’ll suggest paying special attention to the torch-to-work distance. 0.06" is pretty darn close (about the width of a penny, eh? 
) .
It might be good to find a “clean” place on the table, where the stanchions haven’t been buggered, to help ensure the work lies flat (or maybe when the flip happens, hopefully on/around July 15, it’ll work better for this type of cut).
Note there are “fine cut” vs. “low speed fine cut”. The low speed is 1/2 the feedrate of the regular, so you might want to be certain of which consumable and settings you are employing.
Also, if you’re able, use a “edge start” technique, rather than a piercing cut in. Piercing is simply harder on everything, as you have to dump a lot of heat to get started, relative to the amount of heat needed to keep the cut going.
That’s all I can think of at the moment. Hopefully one of our other folks will chime in with your silver bullet. 