I'm a little late to this party, but I burn wood that I cut and stack during the summer. I have a 5' bucket and grapple and have moved a lot of wood and other things with it. In my opinion, the most time efficient method would be as follows:
First of all, unless the stairs under your Bilco are concrete, cover them with a sheet of OSB or something. Your firewood will damage wooden steps and will leave a bunch of semi-rotted and possibly pest infested trash there that will soon rot wooden steps.
If your wood is dumped from a dump truck into a pile, make sure it gets onto pallets or something to allow air to circulate under it. Cover the pile unless you are actively taking wood out of it or you know the weather will be dry.
A grapple will allow you to pick up wood quickly, but the load you get won't be as big as if you stack it into the bucket and use the grapple to help hold it in. I can stack about a bucket and 3/4, then close the grapple to kind of snug it down while I move it. If you just pick it up from a pile with the grapple, you won't carry as much per trip, but you will be able to do more trips in a given time. What's more fun, driving the tractor or stacking wood?
I would seriously consider a carry-all with a pallet on it. If you mess with the design details, you could get a cheap regular carry-all and make a pallet based box with two ends that open. Use the bucket/grapple to load your box, then slide the carry-all under it. Fill the bucket/grapple again and head for the Bilco. Dump the bucket load into the stairwell. Turn around and set the box full of wood off next to the stairwell. Open each end of the box, one facing into the stairwell and one opposite it. Push the wood from the far end into the stairwell with the bucket and you're good to go.
Like I said, you need to mess with the details some so everything fits your tractor and stairwell, but you can move a fairly large amount of wood in a short time this way. I would cover the bottom and sides of the woodbox with OSB to keep things from hanging up, and be sure to make the box wide enough to let the bucket fit inside it.
Depending on your time, talents, and enthusiasm, you can also make up a simple extended toothbar for the bucket to enlarge your bucket capacity.