We have used stoves from the Kifaru Para stove to Kifaru Large box stoves to a variety of home-made designs. They all work best with dry wood, obviously.
The guys I hunt with nearly always carry a saw like the Gerber folding saws or a Wyoming Saw and a small hatchet like the Gerber. A note on the Wyoming Saw blades; newer replacement blades have no 'set' in the teeth. When cutting larger diameter wood the bind terribly and are a frustrating proposition at best.
We have found by cutting larger, dead-standing wood and splitting it, we can spend an hour or so early in the hunt and put up enough wood for our camp for 4-5 days and never worry about wood again. As the wood stacked in the tipi sits, it just gets dryer and burns better as time goes on.
Our policy is to never swing the hatchet or ax in the backcountry. We usually cut a nice 2-3" diameter stick to use as a baton and drive the hatchet thru our wood to split it. That way, there is minimal risk that the ax slips, bounces or glances off and takes a chunk out of a finger, foot, shin, whatever.
Further, at the least the hatchet usually goes in the day-hunt pack in our neck of the woods. The Pacific Northwest is horribly wet as those who live here will attest. The little hatchet can be a real bonus in a bivy situation when you need dry wood for a fire. Much easier to break up and split your way to dry material than trying to find dry enough wood for a fire.