While im not a girl, maybe the girls can verify or disenvow this for me....but if you make EC with a girl you hold it, you smile and you immediatly go over to her...even if your opener is simply "hi". The confidence you show doing that is enough to jump start things. Plus if your that guy standing around with your buddies just staring and making ec without doing anything you'll become creepy which is the worst thing you can ever become.
As for what apraoch is the best...im sure each girl has her preference, but there is NO best way universally...
If you notice a girl and they havent seen you, walk by them if it makes sense to and open over the shoulder. If you make ec then walk straight upto them. If it doesnt make sense to walk past them then you can still walk over to them but over the shoulder wont be right, simply wedge in and open.
One thing i like doing is kinda passive but it gets me opened....thats leading my group within proximity to the group or girl i wana open and then just being the leader and having fun. Also if they dont bite at this point i have a reason to be near them (my friends are here) and i can turn around and open or open over the shoulder now.