I havn't seen any of his videos to know for sure, but I think that Ron Hood gets by with a large knife and he lives in Idaho. Of course, his skill levels are off the chart, so YMMV. Your S1 should be plenty of knife for your needs.
Myself, I would prefer a threesome of edged tools and maybe add a small saw to make it 4. Again, like was mentioned, terrain rules and in a heavily wooded area, I would want my Wetterling 19" limbing axe, which lies between a hatchet and full sized axe. Or, if less heavy chopping is to be done, then maybe a Fiskars 14" hatchet. It will double as a knife if needed, cause it can be made very sharp and will handle all your lite wood working needs. I also like the Fiskars sliding blade saw. Less than 4 ounces and is a heck of a little saw, so why not stow one away in the pack. Under $10.00 at our OSH (hardware). Carry both of the Fiskars tools and you have a great, inexpensive combo that will get the job done. Add a folding knife of some kind and you're set.