We've found nice-sized car kits at local big-box stores. The kit comes in a hard case and is about 8x10 and fits under our seats. We usually stuff in extra bandaids because those always seem to disappear first.
If you have a suitable box (think oreo cookie tins from Christmas time), you can easily build your own. Read a couple of "kit contents" on the ones in the store to get an idea of what to put in your own. Usually cheaper to do as well.
I was going to make a pretty extensive First Aid kit for my 'Field' sports, but I found a Johnson & Johnson First aid kit at Wally-World and supplemented it with some more serious hardware and stored it in an Ammo can.
I like to add things like pre-packaged sutures,
Hemostats, scalpel handle and some sterile scalpel blades,
Eye wash bottle, stuff that the little 'Kits' don't usually have.
Let me see, besides the usual assortment of anti-acids and band-aids,
I have Iodine Solution, probably the best serious wound flush you can have,
I have both Hot & Cold packs,
I have Chem Lights, hard to do first aid in the dark!
Surgical gloves and CPR barrier. You just don't know what people have today...
A big bottle of Tylenol and a healthy tube of anti-biotic ointment,
Topical treatment for poison ivy- sun burns, ect.
A bottle of 'Waterless' anti-microbial hand wash,
Citrus based bug repellant to keep the critters away from a wound...
DON'T FORGET SUN SCREEN! (everyone else will!)
Titanium Oxide based sun screen works best with the least toxic chemicals.
I keep Benadryl in a couple of different forms, good for allergic reactions and bug bites, but will work as a mild sedative.
Also lowers blood pressure for people with heart attacks, stroke or altitude sickness, or just plain flipping out over a situation.
A hand full of the larger 'Telfa' pads so they don't stick to the wounds... They are 4"x4" and a couple of 4"x6".
Usually got a couple of clean garbage bags,
One because I don't like leaving anything in the woods, even first aid packaging,
And TWO, because you can't order the weather.
You may have to dress a serious wound in rain, or try and transport a seriously wounded person out where they are going to get the wound muddy, wet, exposed to insects or sun burn...
Anyway, to keep it all dry and safe when driving, I keep it in an old Ammo Can (repainted with the 'Red Cross'),
When I'm Hiking, I just take the smaller plastic box.
Look in the Trauma Kits area. We have vehicle specific and general trauma kits of all sizes, but the smaller ones are all designed to fit in a BDU pocket. All you're going to get at a big-box store is a giant booboo kit. Their idea of wound care beyond the Band-Aid is a couple packs of square gauze and a roll gauze to tie it on with, even in the biggest of them. You get much better results from even a single 4in "Israeli"/Emergency Bandage (see toward the top of the OP), like the ability to apply it with a single hand for example.
If you want to use your own container, this link shows just the supply refill for a first responder BLS kit. If you have a higher level of training or need, the EMS section has a full spectrum of individual items as well as prepared first aid and emergency response kits and supplies, cervical collars, backboards, stokes litters, SAM splints, clotting agents, burn dressings, etc.