You are doing a very nice job on your quiver. It is difficult to keep the tension right so you can control the shape. I like your hourglass shape, intentional or not!
Honeysuckle vine works well but I think you have to strip the bark and boil the vines. Grape vines work but can be a little bulky. If you live in the land of palms and palmettos there is lots of good weavers there. Jute cordage, like you have in the bottom will work well all the way up or as accent strips that will give the quiver more durability.
A bit later in the spring the inner bark from some trees is quite strong and would work well too. This outter bark from some tree saplings might work as well.