@jess, I don't think that the material is Cordura. My guess that it is pack cloth that was reinforced. (See @adalangdon's response in
this thread). One solution that I posted up on Ravelry, but which doesn't seem to have gotten a lot of attention, is to use the neoprene handle wraps @celtical originally recommended that
were mentioned recently in this post.
The
COSMOS neoprene padded handle wraps are short enough at 5.3”/275 mm to work as padded grab handle wraps for the Cafe Bags, Synapses, Daylight Backpacks, etc. as well as for the webbing handles of the Daylight Briefcase. I’ll add that this will also work as a strap wrap for the Swift, although I’ve never had to use one for my Swift or Little Swift.

Here’s a picture of the COSMOS neoprene padded handle wrap around a Swift, with the unwrapped handle wrap shown to the right.

Demo photos showing that this wrap is soft and flexible enough to work with many other webbing handle loop uses -- such as for a Synapse or for a Field Journal Notebook. Again, an unwrapped handle wrap is shown in the upper right. It’s a little longer than the length of a Mini organizer pouch or clear organizer wallet. A size #4 Knitting Tool Pouch is slightly larger, but better approximates the unwrapped size.
On the Swift, it’s the fact that the padded portions of the handle are under the wrap that makes the wrapped section look straight -- not the fact that the handle wrap lacks softness and flexibility. The usual issue with finding a good handle wrap to use for these purposes is that they’re too long and/or too stiff to work well. So, while I can’t give you a “feel” of the wrap in these pictures, I can show instances where it is really flexible.
If you read the original post I linked, there are links to other colors of handle wraps, or you can search for them yourself. Black should work for most purposes.
HTH
moriond