I define "emergency" as an immediate threat to life. limb, or eyesight.
As such, my "emergency" first aid kid is essentially an IFAK kit. Those with prior military experience are probably familiar the the concept of a Individual First Aid Kit. Some years ago the Dept. of Defense took an in-depth look at what people were actually dying from in combat and then starting pushing these IFAK kits to address those immediate, life threatening injuries. The majority of combat related deaths were identified to be a result of essentially two things, uncontrolled bleeding and an inability to breath due to pneumo/hemo thorax conditions. The military designed and then began issuing IFAKs to troops combined with training on how to use the contents. Combat deaths dropped dramatically as a result. The idea behind an IFAK kit is that it is for you to treat you.
The contents are fairly simple and again are aimed at dealing with the most immediate threats to life, i.e. supplies to control bleeding, alleviate a pneumothorax, and establish+maintain an airway. A basic IFAK typically includes, at a minimum, the following:
-Chest seal x2
-Tourniquet x1
-Hemostatic pressure bandage x1
-Hemostatic gauze x2
-trauma shears x1
-nitrile gloves x1
-chest decompression needle x2
-cpr mask (numask preferred)
-NP airway x2
Combined with training, these supplies should allow you to address the most immediate life threats, that is to control life threatening bleeding and establish and maintain an airway. My IFAK kits goes with me on a daily basis whether in the car or out for a day hike. Some stronger pain meds would be a nice addition if you have that option. Although not a lifesaving intervention, they do a long way with regard to patient care and comfort should you or a buddy get badly injured. This kit can be ultra compact. Mine weighs about 10oz and fits in a 6x6 pouch.
For remote backpack or fly-in hunts where definitive care is more than an hour away, the discussion is more about short to long-term care as opposed to "emergency management" in my opinion. The IFAK supplies deal with the "emergency" and the rest of the kit manages short or long-term care as you and your team move towards a higher standard of care.
If I am more than an hour from help and/or on a remote backpack or fly-in hunt, the basic stuff described above stays, but I consider adding the following:
-More wound management stuff like bandages, gauze, butterfly closures, skin glue, etc.
-Consider injured joints and limbs and add things like ace wraps, SAM splints, sling and swathe materials, etc.
-Blister and burn management supplies
-basic meds like ibu, Imodium, and infection prevention/antibiotic ointment, etc.
I used to carry custom first aid kits based on my personal preferences, training and experience, but lately I've been using the kits by Adventure Medical. They have a variety of offerings from basic and pretty sparse to super comprehensive. I usually add a few things to these kits to set them up just as I want.
Far more important than any piece of med kit you bring along is the knowledge, training, and experience that should go with the proper application of that piece of gear. At a minimum a three-day Wilderness First Aid type course should be considered, if not a week to ten-day Wilderness First Responder course. I would prefer to have a partner with more training as opposed to a partner with a bigger first aid kit any day of the week.
Good luck!