If you do much machining you will have several tools with Morse tapers, either for use in a milling machine (without a tang so a draw bar can screw into the end of the taper to hold it in place), or for use in a lathe, to usually be held in the tailstock, (in which case it will have a tang).
MT3 without a tang
MT3 with a homemade tang added. The purpose of the tang is to prevent the taper from turning in a lathe tailstock, but this taper cannot be held securely enough for use in a mill.
So you will need the "tang" part too screw into the thread of the tang-less taper.
And it's a very simple matter to make two like this whilst you have the work set up on your mill, then simply cut them in half. This idea will work for any size Morse taper, simply use a taper with a manufactured tang to get the dimensions.
You will then have the best of both worlds, a Morse taper tool to use in both a lathe and mill. Make the male thread size, pitch and length to suit your tooling and unscrew the tang whenever necessary.
Hope you also benefit from this idea.