Note Well: I am not a medical professional and all advice I give about this should be independently verified.

The first thing is to update and hopefully upgrade your first aid training. The kit between your ears matters far more than the kit in your pack.

Gloves should be in your kit. You can do first aid without them, but they give you much more peace of mind when working on strangers.

You should have a watch for reading the pulse rate and a notebook with a pen to write it down with.

A cigar box should be big enough for a pair of crash scissors (bandage scissors?).
If you need to cut clothing they are less scary than a knife.

Sticky band-aids are good for minor cuts.
Include a few aspirins and something sweet. Aspirin is for pain, but also for heart attack victims. At the first symptoms of a heart attack the victim should chew one tablet of aspirin for 1/2 a minute then swallow it. You can use Alka-Seltzer tablets the same way.
Candy or sugar can be given to diabetics if they are going into shock. If they are low on sugar it will help them, and if they have too much sugar it will not hurt them.

You should also have a charged cell phone.
If you regularly carry a cell phone then it is taken care of, but if you don't then you can get an old one just for emergency use. It does not need to have any minutes on it. It does not even need to have an account. So long as it is charged you can dial 911. Even if you have a cord to plug it into a car lighter socket or a wall charger it is fine.
I would not try carrying a lot of bulky bandage material, there is usually more than enough stuff around to use for bandages, padding, slings or splints. A small roll of adhesive tape might be handy.


Remember that you are just doing first aid. If it is at all serious they need to get proper medical attention as soon as possible. Since it is an urban kit this is easier.
So you do not need to spend time cleaning wounds. The hospital will do that and load them up with whatever drugs they think are right too, so no antibiotics, heavy painkillers or even wound cleaning if you can help it.
That eliminates a lot of stuff.

If you (or maybe your family members) have a problem like allergies and need an EpiPen it should be in your kit. The same for any other special meds they might need.
A small pair of tweezers would be good for splinters, but don't dig or cut to get a splinter out. Splinters usually just pus up and come out on their own.