You might as well be ready to take care of strained knees and ankles. I think the single most useful tool in that endeavor is a large elastic bandage or two. The four inch width is not necessarily too big for ankles. If it is, a tight ski boot is probably better support for most ankle injuries than an ace bandage, and certainly warmer. So it seems to me you should carry the bigger size.
Elastic bandages can play several different roles in an athletic injury stabilization effort.
Wrapped around the damaged joint, they prevent swelling and limit movement somewhat. Early prevention of swelling is important to the long term recovery of the joint. After the joint is compressed, it should be braced with some kind of splint. A second ace bandage is often the easiest way to hold the splint in place because its elasticity brings the limb, the splinting material, and cushioning together nicely. Especially during cold weather, it is important that extremities continue to receive adequate blood flow.