Thanks for posting. Yes, the new command we are envisioning would be helpful here.
For now, the Points command has an offset set to the point variable by default. You can change that here and type in a pixel offset to move the labels.
That said, the offset will only increase in the same direction when using the Points.
One thing you might want to try is adding the labels using a Plot command, instead of Points. You would add the additional Plot command without drawing the line. The benefit here is that the default location of the labels is set to avoid where the line would be so that some labels will be up and some down.
You can increase the Label offset and change the background to white.
Again, not a perfect solution, but if you use that approach most of the labels may be OK. For those that are still overlapping, you may need to use a Label command to draw an individual label with an arrow to the point instead of having the label in the data table.
Hope this helps for now!