If you take care of your axe and maintain the quality of the wood by oiling it and protecting it from the elements, you shouldn’t have to deal with this problem with any regularity, if ever. Things however don’t always work out in the best possible way, and sometimes, either through our errors, or because of unavoidable circumstances, our tools get damaged.
One of the most common issues with axes which utilize wood handles is the loosening of the axe head from the handle due to shrinking of the wood. If that happens, the axe becomes a very dangerous tool, as the head can potentially fly off during a swing.
There are two ways to fix this. One is to replace the handle. Another is to replace the wedge at the top end of the handle. Many axes have a wooden wedge in the eye of the head, which stretches out the handle and provides for a secure fit.
In an emergency however, there is a much quicker fix. If you have a nail or two handy, simply hammer them into the center of the handle through the eye. If the nails are thick enough, they will spread out the wood enough to provide for a good grip.
When I was younger, that was the way we typically fixed all of our axes. The method works, but remember, just because you have hammered in a nail means nothing if the head is still loose. You may need a thinker nail, or the addition of another nail. I wouldn’t use more than two or three nails, as it might completely damage the handle. Also, make sure the nails are not longer than the eye of the head. You wouldn’t want them in the part of the handle below the axe head. Short, thick nail are best.