As to needing a fresh base layer daily...we are all different, but I would like to mention something that I discovered several years ago and like very much:
http://www.luckyvitamin.com/086449300093.html
This "crystal" spray deodorant is really wonderful. It has no odor at all, and seems like plain water. What it actually is is a solution of mineral salts in water. It does NOT have aluminum salts, but has other mineral salts like perhaps magnesium salts, etc that are harmless. How it works is that it creates a PH environment on your skin that the underarm bacteria cannot thrive in. Odor happens when those bacteria multiply over the day. The mineral salts create a pH that the bacteria have a hard time breeding in. So in a way this deodorant works the very same way as the wool does- creating an underarm environment where bacteria will not thrive and multiply much.
I really like this stuff and it is much more natural than most commercial deodorants. It leaves no residue and does not have any perfumes at all.
It also comes in a crystal salt "rock" form that you wet and glide on, but I like the little pump spray myself. (non-aerosol of course, so doesn't hurt the environment) :)