It is possibly getting over stretched (lengthways) when machining it on. Or not stretched enough, or is being pulled in the wrong direction ( and too much) when hand sewing. On consideration I think most likely pulled too tight and also it has rotated slightly on the second line of sewing - the hand stitching, this can be very easy to do, just pushing the fabric along a bit too far with each stitch so it ends up twisting the whole thing. Try unpicking the hand stitching and then re stitching but going in the opposite direction for a few inches to see if this helps.
Most of this is musing out aloud, it can be tricky to work out a problem without seeing it, but I hope my musings help you analyse where/how it is going wrong.