Mom of 4 here - We use Schmidt's as mentioned by someone else - love it. Wal*Mart has some generic brands similar to Tom's that we use & like. All depends on how strict you are when you say "natural". My Italian husband & I (mentioned italian bc he S.W.E.A.T.S.) went completely natural - as in NO deodorant for about 2 years. There was DEFINITELY a detox stage for me where my body shed lots of harboured toxins and I smelled like a buzzard. But then things evened out and I just had to shower daily to stay fresh. I eventually started using Schmidt's bc whilst pregnant. Anyway, my husband still wears nothing and smells fine! So, short answer: Schmidt's, Wal-Mart generic of Tom's or NOTHING.
@Chumly I don’t think the author was asking specifically for backpacking, just being active outdoors in general. I’m a backpacker and although I don’t wear deodorant at all while doing that activity, there are plenty of other outdoor activities where I would wear deodorant and I think that’s what the author was asking.
It is good to note thought that yes, toiletries fragrances will attract bears, so when camping overnight somewhere in bear country, the practice of wearing smelly stuff and not storing it in a bear-proof container is discouraged.
I use a brand called N-O-N-E. I agree with @Chumly that any fragrance can and will attract critters, small and large. Car camping in a state park is one thing, but backpacking miles into the backcountry is another. Yes, I can get a tad ripe after several days in the summertime, but I carry a small, quick-dry pack cloth so I can take a quick, but effective, stream bath. A vigorous scrub of the 'pits with the wet cloth and I'm good for a few more days. Plus if you're backpacking there's a few ounces saved!
At Philmont Scout Ranch, deoderant of any kind is absolutely forbidden on the trail - not even allowed in your pack. For a good reason IMO.