Having a rope along for the scrambly day hikes is a good thing. Beyond that, I suggest that day hikers carry the 10 essentials, per the Seattle Mountaineers:
  1. Navigation: Whatever it takes to find your way back if you are separated from the group.
  2. Sun Protection: Includes hat, sunscreen, sun glasses, etc.  
  3. Insulation:  Basically extra clothing. Your insulation should allow you to survive the worst conditions that can be realistically expected.
  4. Illumination:  Whatever light you need if you are delayed and must travel after dark.
  5. First Aid Supplies
  6. Fire: Matches or lighters
  7. Repair Kit and Tools:  Knife or multitool. And duct tape, perhaps wrapped around your poles.
  8. Nutrition:  Something you won't normally eat, to hold in reserve.
  9. Hydration: Enough water for the day. Purification will help in a pinch.
  10. Emergency Shelter:  To keep you warm if you are stuck overnight - space blanket or an extra-large plastic trash bag
It is good to have a cell phone or two along. And bicyclists need spare tubes and tools.