I had this with both my babies and I would put a folded burp cloth (I have the baby diaper types Gerber used to make, with the thick part in the middle) under their heads, under the jaw along the neck and out on top of their chest (can you visualize this?) It provided a cushion and a place for all that extra milk to go (instead of on me) when she let go and needed a breather.
They both learned very quickly how to eat most effectively, and we made sure to burp thoroughly after every feeding. I also try to burp her BEFORE every feeding for a minute or two, even now. If your little one is spitting up a lot, are you keeping her elevated after feeding (like a bouncy seat?), but don't bend her in her middle (like sitting her on your lap.) I thought it was neat that by the time my latest was 4 weeks old (she's 7 months now), she would calm down from crying while I was putting the cloth under her chin - she had learned that it meant the food was coming! = )
The idea of pumping a bit beforehand sounds good, it didn't work for me because I needed the baby to help with letdown. If it works for you, I would try that, but my trick helps if you ever have to feed while you're out or don't have time to pump right before. At least then you won't be wearing it!
Good luck!