One of the biggest things you need to accept to save fabric is that the factory folded edge isn't the only folded edge a piece of fabric can have. You can refold it however you want!
This is just an example. With my raglan top pattern, if I used traditional fabric layout, I'd need over a yard of fabric to make a toddler top. That's just crazy. Plus, I'd end up with a weird piece, cut off from the folded edge. What would I do with it? Instead, I folded over the raw edge and the folded edge to make 2 double folded edges.
Now I can cut out 2 sleeves (on the fold) and the front and back (on the fold) from 1/2 yard of fabric. Even If I only needed to cut out a sleeve, I'd still use this technique and fold over the raw edge. Then, I could cut out my 2 sleeve at once on the fold and have a nice, large piece of my fabric left with the factory folded edge to use on another project.