A running Weimaraner may not need (or want) a coat; however, their feet might be at risk. Ice melter (and salt) can be toxic. This is true even if it is dog-safe; it may still wreak havoc on their paws. Consider booties or apply a special waxy protective coating.
One well-known product is Musher's Secret. Here is a short explanation from the makers of Musher's Secret about their product....
Safe and Natural
paw protection; or paws for protection :O)