Bows were unstrung and wrapped when not in use. This isn’t completely true of modern bows, but was the case with most historical bows.
The wrapped bow could be carried over the shoulder or attached to a pack, but it’s still a fairly sensitive item. It would also need to be kept dry. This was primarily because the glues were water soluble.
Your character probably wouldn’t want to keep the wrapped bow near them in combat, and if they’re dumping their pack somewhere before a fight, then that’s where their bow would be.