To help accomplish that we can borrow from the machine shop world. That would be a center finder.
A center finder is a hardened steel cylinder and the tip is held on with a tension spring.
To use it, put it in a drill chuck. Push the tip off center and turn on the drill press at a low speed between 500 and 1000 RPM. You don't need any more than that.
With the chuck spinning, put slight pressure against the side of the tip until the wobble stops. I once worked with a guy that used his finger nail - he had 9 1/2 fingers. Use a screwdriver.
Now the point is exactly on center with the axis of the spindle. Use it to align with your center punch mark or crosshairs.
If you look closely, you can see that the first attempt is a bit off center. I fixed that with a center finder.
If you look closely, you can see that the first attempt is a bit off center. I fixed that with a center finder.