Should I start by tossing some dried leaves and twigs in the run?
You could put some leaves and twigs in there for now, don't make it real deep or the leaves will just compact down(not good). I would get some aged tree trimming chips or undyed mulch in there asap also. This will give you a good start for semi/deep litter. Once you get the chips or mulch, you can add your yard and garden waste. Wood chips/ mulch helps water drain down through the litter. Think of a mulched flower bed with extra stuff in it.
Here's my run after adding yard and garden waste, before the chickens got to it.
They like to dust bath in it also.