Greg is correct that the satellite antenna prefers to be flat side up. Hanging vertically like in the photo is not too bad, although you might see a bit of loss of accuracy for position, especially vertical position. It might take the sat receiver longer to lock if you hang it in a very bad orientation. I would not expect it to crap out completely. If you face the antenna directly toward a human body, you will get more loss of accuracy and loss of lock.
I know this is not easy, but the best way to get accuracy is to hold it in your palm with an outstretched arm so that it is fully away from most of your body. The signals will go through all fabrics except for fabric with metal threads included. It makes it a bit harder if the fabric has snow piled up over it, since water tends to absorb the signals. Mesh pockets tend to work good. A ziplock plastic bag will work as a holder.
I've seen some people go so far as to place the receiver flat side up on a sheet of metal, which acts as a ground plane. I think that might be overkill.
–B.G.–