Just found this thread and have been wondering about the same issues, as I started using a Jenkins-inspired system last Fall. Have tried medium-small leaves, but the problem of Fall moisture in the Seattle area does make them a bit stinky, not to mention buggy. Just couldn't get them dry enough outside in Fall, even under cover, although they were better than nothing. Tried also fairly dry fallen needles that I had gathered in summer from my Hemlock tree. Those were pretty stinky, too, when they were "moistened" in my bucket loo. Ash from the fireplace has been the best so far in terms of odor, but it's kind of impossible to avoid breathing in some of the dust as I sprinkle it on, plus I generate ash only seasonally. Have also tried sawdust generated by moistening fireplace pellets--admittedly a commercial solution--and it works OK with the odor but not as well as the ash. Recently kept some self-generated sawdust from sawing up some fallen tree limbs due to recent snows. That sawdust sat in sun for a while and was pretty dry, and worked well. Coffee grounds--even moist ones-- work well too for loo odor, but it takes at least a few day to generate enough of them in my family to cover even one solid-waste "event". I should try saving grass clippings from mowings this summer, as some posters suggest, as well as just digging some soil (after it's dried out a bit here).
Overall, I've learned that you can use a variety of what's available, and that some things work better than others. I really like the idea of just digging soil out of the yard and trying it for loo cover material, because when my compost piles have matured, I'm going to need some gaps in the yard to put the compost in!
BTW I came across this youtube vid (https://www.youtube.com/watch?v=bUzTVs8iDPE
) that shows an "ash flush toilet" from perhaps 50+ years ago. It's basically just a hopper for ash that might have very conveniently covered your business in its heyday. Might have worked with sawdust too. Seems like basically a large "bulk bin" type hopper mechanism, like those in some supermarkets that deliver bulk flours into plastic bags, except with a pull chain to dispense it.