Not sure if there is a way to get Notes to do that, but you may be able to find another note taking program that can be assigned to a button and have it start out with a new note.
There are some free and commercial note taking programs you might be interested in. There are also some that let you take notes right on the today screen, so you wouldn't have to press any buttons.
Take a look at freewareppc.com, handango.com, and pocketgear.com.
There is also this program that is free. It has a calc and a note program in it.
http://freewareppc.com/utilities/calcnote.shtml
I beleieve Pocket Max is a Today plugin that you can purchase that will allow you to, among other things, take notes on your Today screen.