A blanket or towel inside would be nice, keep the little bird warmer and offers a little cushion for it so it may help prevent it from sliding around too much As was said, the bird should be used to a travel carrier or box before hand if possible and should be kept out of drafts in the car. Try to make it so that there are no drastic temperature changes. Make sure the box or carrier is secure, have someone hold it or something. Keeping the noises to a minimum is probably best to keep the bird less stressed.
By the way, it's so, so cute!!