Tools.

Discussing tools with David Chubb (His website is called Cotswold Beeskeps) I discovered that he uses a much narrower fid than mine, made from automobile brake pipe and used in the opposite way.  Where I push my fid through the straw in the same direction as the binder, David pushes his fid through from the other side, inserts the cane and pulls it through with the fid. David makes many more skeps than I do and his technique obviously works well.
Recently I have experimented with use of a spike rather than a fid. I made the spikes as awls for willow basket work by grinding down some old thick screwdrivers to make tapered points, as shown here.
Do not carry one of these in public - they are dangerous weapons in the wrong hands. I keep a wine cork on each one for safety when not in use.

Sharpened screwdriver and a home-made cleave

The method is simple: pierce the straw with the spike, pull the spike out and immediately push the cane through the hole before the straw closes up again. It helps if you sharpen the end of he cane a little.
It's a good method for working in tight corners and probably just as quick as using a fid, with less pushing effort. However, there is the risk of spiking a hole in your other hand or leg, not possible with a fid.  And without a third hand there is quite a lot of putting down and picking up again for each new stitch.
Somewhere on the web you can find photos of George Hawthorne (sadly long gone) using what looks like very large nail to make the holes in the straw. There are also pictures of continental beekeepers using spikes in the same way.

Materials.

Using split willow as a binder has its merits. It is fairly easy where I live to find willow sticks, this year's growth. Choose long unbranched shoots a little thicker than a pencil. Cut them in the autumn or winter and split them with a cleave. The cleave splits the willow into three long pieces (called skeins by basketmakers) and working with these fresh gives a nice pattern of green or brown bark on the skep.

Here is the cleave splitting a freshly cut willow stick.

A cleave is a hard piece of wood about 5 cm thick, cut with a saw and carefully trimmed with a knife to give three sharp edges meeting at the centre of one end. The willow is opened up a little into three, using a knife, at the thin end (first cut off the very thin useless tip) and the cleave is pushed through ithe stick to split it end to end.  Scrape off the pith from the willow and sharpen the thick end before use so that it is easy to push through the straw. It is possible to use the sharp end of the willow to make the holes and do without a fid or a spike much of the time.

Fresh willow skeins used to bind the start of a new skep

If you leave it until the spring to cut your willow sticks they will have started preparations for growth and the bark will peel off very easily. This is a nuisance if you are trying to make something like the photo above, but it also allows you to strip the bark off leaving white wood which can be softened up and used just like strips of cane or branble.
I am trying to think of a good use for all the long leathery waste strips of bark this produces - probably something decorative in willow basketry rather than straw skeps.
A skep might look more decorative with alternating bands of different colours of binder. No more useful, but more decorative.

Wooden additions.

There are photos of museum specimen skeps that have square wooden frames at the open end so that they would sit on a modern hive. I am not sure whether to start the straw work at the edge of the box and finish at the top of the skep, or to start in the normal way and build out to a square that can be sewn onto the wooden frame. Probably the latter. Any advice on this?