I warped up the little loom again last week, thinking about this problem as I did so. I wind the warp around the loom first, and then put the heddles on (yellow threads in the picture below). The yellow heddles are restraining alternate threads, the un-restrained threads can then be raised and lowered by hand to create the shed to pass the weft thread through.
Putting these heddles on was creating uneven tension - why didn't I realise that before? The warp is a continuous loop - when I change colour I tie a knot and keep wrapping the thread around the loom. This means that uneven tension can be cured by working the warp about and moving it around the loom so the stresses settle out.
It worked perfectly. The picture above shows the first few inches of band I wove. I am so pleased to have fixed this little problem, all my inkle bands can be beautiful now. I could think about some more fancy weaving patterns. There are some lovely, more complex, designs that can be woven using pick up techniques. See the work of Tracy DeGarmo, Heather Heroldt, and Sara Lamb.