Typically what I want to do to just barely break through is define my stock as slightly thicker than it actually is. E.g. for a 0.25 inch thick stock, I’ll make the depth of cut (or thickness of stock) around 0.26 inch depending on how flat i trust my stock to be. Since you say you have a 0.01 inch or 0.02 inch thin skin, i’d suggest making your stock 0.02 inches thicker.
You can also just set the z zero slightly lower as long as your bit, feeds & speeds can handle cutting the additional material in the first pass.
Now if your only doing 1 sided carving this is great, but when you get into flipping things you could end up overcutting the middle.
Also, I’ve found measurements need to be pretty exact… meaning even though it’s 1/4 inch MDF, it may actually be 0.26 inches thick. so if I specify 0.25 depth of cut (stock thickness) i’m left with 0.01 inches of material.
Is the paper thin skin consistent around the entire part? if not your spoil board may not be ‘level’ to the x/y axis. Search ‘surface waste board’ or ‘surface spoil board’ for more information.