With a newborn I think the Moby is really the best poppable carrier there is. Tie it on and leave it on throughout the day, no need to retie to put baby in and out. For a super fast carry with a newborn I like a ring sling a lot as well, obviously super poppable, but I like a two shouldered carry as they start to get bigger.
A woven wrap could be used to free up your hands with a SHBC, which I love when I am chasing a little one, having the baby safe and on my back is sometimes easier than on my front. I use a SHBC a lot for naps with teeny ones, but I was so scared to try it at first it took me a while to learn. :)
The beco Gemini is supposed to be good for little babies, and a mei tai would be another option to consider as you can scrunchie the bottom to cinch it downwhile the babe is small, and still use it as they grow.
What is it about the Moby that isn't fast enough for you?