If there's one thing you learn when hiking, it's that accidents happen. Here's the essential list of first aid items you should have with you on every hike.
I used to be strongly against relying on a cellphone - and certainly it's not always going to work. But carry it for those situations where you're within range. It's an important back-up for when serious injuries occur. Also carry whistles. Those can alert others within hearing range to the problem.
Band-aids and Bandages
You don't need a thousand of them. Just a few to cover most common scrapes and injuries. The bandage can serve double-duty for other tasks - but it's good to have one or two fabric strips as part of your gear. Just in case.
You should always have water with you anyway, but it's important for first aid reasons, for washing out wounds.
There can be all sorts of dangerous stuff in wounds in the woods. Rusty nails. Animal feces. It's important to treat it quickly and then get out to a trained medical location for further care.
Knife and Tweezers
It's a good idea to carry a good multi-tool with you on hikes. You never know what might happen. You could get snared on an old fishing hook.
It can help to take off the edge until you get home again.
Bee Sting Treatment
All sorts of bugs hang out in the woods. Be prepared for a less-than-friendly encounter with one.
There are of course all manner of pre-made kits out there and it might be cheaper to simply buy one of those. Make sure it has the basics of what you need. You'll probably have to get the special items like the multi-purpose knife separately.
Also, even if you don't have cell phone coverage where you'll be, load up your phone with important data and bring a battery backup for it. Things like photos of poisonous plants and animals and how to handle encounters with both. A first-aid guide can be very handy as well. The more data you have with you, the more you'll be able to handle whatever you run into.
Finally, invest the time into a first-aid course and CPR course. I grew up as a Girl Scout and believe strongly in the Be Prepared motto. I worked with the Safety Committee at many of the biotechs I worked for. You never know what bizarre accident might happen. The more you are trained to respond, the better the outcome will be.