Okay, the 1/8" end mill as a boring bar worked well to square up the bottom of the hole and bring the diameter to 0.374"
Now the silly story:
I got some 3/8" diameter neodymium magnets from eBay a while ago and made a bunch of really strong refrigerator magnets. They hold lots of things and have a nice handle.


The body is aluminum and the magnet just pressed into a 0.374" hole.
Then I read on a machinist's website that he increased the pulling power of his magnet by wrapping the one pole around to the front. Like a horseshoe magnet, but where one pole surrounds the other. At least that was his contention.
My setup is like this:
- An aluminum body just like my original fridge magnet.
- The business end is bored out 5/8" diameter to a depth of 3/8" (or so).
- A plug of mild steel is pressed into this hole and the whole thing faced off.
- The steel insert is bored out 1/2" diameter to a depth of 1/4" (the height of the cylindrical magnet).
- A plug of aluminum is pressed into this hole and the whole thing faced off.
- The aluminum insert is bored out 0.374" to a depth of 1/4" to expose the steel face.
- The neodymium magnet is pressed into this final hole and the back face (and only the back face) comes into contact with the steel insert.
This picture shows that one magnet pole comes directly from the circular face of the magnet and the other pole should be coming from the steel ring, separated by the aluminum.

The proposition is that instead of wasting the pulling power of one end of the magnet its lines of force are made to be redirected out front so that both N and S poles can be used.
I have not tested it with lifting a filling bucket of water to see how much it can lift versus the regular fridge magnet. But the seat of the pants test against the fridge says that both versions have the same pulling power even though a piece of ferrous metal is attracted to both the magnet face and the steel ring.