Best weapon for game-getting that will fit in PSK / EDC type usage is, without a doubt, the ancient sling. If you carry twine and ranger-bands you have one already. Only trouble with this weapon is the required practice to become proficient with it. With practice it is possible to take game up to deer with them from quite a distance. The ammo is lying at your feet everywhere.

Required practice time to learn and become proficient is around 2 hours a day for two months or so. If you have a natural talent for these sort of things it may go quicker.

The next easily carried EDC weapon for hunting would be the improvised elastic slingshot. If you are carrying twine and some ranger bands or some latex tubing you can improvise one from any stout Y shaped branch you could pick up.

You could also carry the flat hunting arrow heads and fab a spear or atl-atl. Of the two of these the atl-atl is the more deadly / longer range tool but requires better training and practice to become skillful.