The official Brymen cases have a decent felt liner and the EEVBlog case has a thin layer of padding between the inner and outer layers of fabric. Both provide a reasonable amount of protection to the DMM and are generally designed to allow the leads to be folded up to the left or right of the DMM while stored in the case. Welectron has both cases listed, but I wonder if the EEVBlog case has been discontinued... Dave?
Neither are "single layer paper thin", but of course are also not quite comparable to an EVA case. There are tons of EVA cases available all over the 'net, so just start searching for something with the right dimensions. You can cut some foam in the shape of the DMM and the accessories.