Fast Alternate Palm Muting?

Hi all, first time poster here with a question. How in the world do you play really fast palm muted chords? I can play palm muted stuff decently fast, but when I try to do it with upstrokes AND downstrokes I can't really angle the pick up enough or get enough leverage to get it to sound good. I've heard some talk about holding the pick at an sideways angle, and this makes it a lot easier but it sounds much worse.

I'm not sure what the rules about copyrighted material are on this board, but would a short excerpt from a song just to illustrate what I'm talking about be appropriate?