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If you have a hammer or axe handle with a broken-off handle in it, the first thing you need to do is get the remainder of the old handle out. I normally do that by letting any handle stub hang between the jaws of my heaviest vice, so it just barely has room to move. Then I drive the old handle out using an old bolt and a heavy hammer. PLEASE, PLEASE, PLEASE do not follow the advice that I’ve read a time or two and throw the head in a campfire to burn the handle out. That removes the temper and RUINS the tool, unless you’re an expert at retempering.
No doubt, there are better ways to get a handle (haft) in place, but after driving a handle in an axe or hammer head until it’s very tight, I simply use a half-round file to make a very slight groove around the handle, right against the head; then I drive the handle in a bit further. I keep repeating this until the handle extends beyond the tool head the desired distance (1/8” to 1/2”) and then wedge it in place. On picks and such, you pretty much have to drive in the handle (you can do this by thumping it on the ground, or floor), then take the handle back out. Look for marks where the head is tight on the handle and rasp those places down a little. Then you keep repeating the process until the head is where you want it on the handle. You then cut the extension off ¾” or more beyond the handle. (I left about 1-1/4” on my last one) Finish by rasping the edges smooth, so they’ll be less likely to splinter from being thumped around.
I must confess that I’ve only made a handful of handles in my life. I found that I could buy handles cheaply enough that it never paid me to take the time. (That may be changing, now that I’m a poverty-stricken retiree.) Of course, since handles are now made from sawn lumber, rather than split blanks, you really have to inspect the grain carefully before you buy. The least bit of run-out in the grain may cause breakage in the future, especially if the cross-grain is in the half of the handle towards the head. I prefer the grain to run from front to back in the handle, though I HAVE read of folks who prefer side to side. The only problem with front to back grain is that the handle may bow, if left out in the weather. The solution to THAT problem should be obvious.
On those tools where the handle is inserted from the top, like adzes, picks, mattocks, eye hoes, tomahawks, etc., the extension is left above the head to allow for wear and wood shrinkage. If the handle is 100% dry, it may not shrink. However, if you keep the tool for years and use it a lot, there will be some wear, since it isn’t wedged in place, but is only a friction fit. If you were to be using a pick or mattock for old-fashioned excavating, where you would be working a vertical face higher than the blade is long, you may not be able to leave as quite as much handle extended beyond the head, as it may hit the “wall” of the hole. If you’re only working blade deep, as in a flower bed, you could probably get by with more.
A lot of old picks and mattocks that I’ve seen have the protruding handle worn into a dome shape, rather than the straight cut as it began. This is usually from hitting the wall, but is also caused by storing the tool by standing it on dirt or concrete. That allows moisture to wick into the handle and rot off a bit at a time. The same thing can happen with axes and sledge hammers and such. I’ve seen tools where the handle was sound, but the wood in the eye of the tool was too rotten to allow the tool to be used. To prevent that problem, either hang the tool up when not in use, or put a brick or something under the end of the head so the handle stub won’t come in contact with the ground or concrete. It’s best to put something water-proof, like a piece of asphalt shingle atop the brick, too. Naturally, if you store the tool in a building with an above ground wooden floor, there is no concern at all. Incidentally, I’ve seen some really old tools that had shims of tin, rawhide, or leather in the eye to make an otherwise worn-out handle serve a while longer.
On tools that have a handle that fits in the head from the bottom (axes, hammers, etc.), little extension is needed. Since the eyes of those tools normally have a double taper, they are wedged in place. When installing the handle, I’ve learned to take a file and chamfer the top of the wedge slot a little, which makes it easier to start the wedge, once the handle is driven in place. After the wooden wedge is firmly in place, I trim the stub to the desired length. If there’s a gap front to back in the eye, use the little metal wedges made for that purpose and drive them in cross-wise of the wooden wedge. They can usually be placed near the center in a hammer handle. On an axe handle, I use two if any are needed, and place one about an inch from each narrow edge of the handle. If the handle shrinks a little more with time, just drive the handle in deeper and drive the wedge(s) in further.