A new handle can be made from a piece of metal tube or a wooden dowel.
Choose a new handle that has a diameter the same size as the internal diameter of the T-joint.
For a strong wooden handle, ensure the grain of the wood runs along the entire length of the handle. Grains that run across the handle weaken the handle because they make it more likely to snap.
Treat the new handle before you place it through the eye. Wooden handles should be treated with preservative and metal with paint or a thin layer of oil.
Metal handles should slide in easily or with the help of a small amount of lubricant such as silicone spray.
Choosing wooden dowel with a diameter slightly larger than needed is fine because you can use sandpaper to sand down the handle to a perfect fit. If necessary, clamp the manual post-hole auger in a vice and use a wooden mallet as described in How to remove a manual post-hole auger handle to hammer the handle through the eye.
Hand grips can be added to the handles once they are in place. The hand grips will stop the handles sliding out of the T-joint if the handle is loose.
Holes can be drilled in the T-joint and handle so that bolts can be used to hold the handle in place, although this is not essential.