First off I am a little confused could you clarify if this is correct. If you have a layer that disappears at 1:24000 and you have a label that disappears at 1:10000 , how could you ever have an extent where the layer disappears but the label still appears (ie to disappear it has to be greater than 1:24000 and that would make the label disappear since its at 1:10000) I could be just misunderstanding what you mean.
Two ways I know of. One which is probably the easiest is to just create a second copy of that layer. This layer you will symbolize with nothing but turn the labels on. That will only display labels. You can then set them to appear at the extent you wish.
The second way would be to convert the labels to annotation. You can then display the labels at different extents than the objects.
Arc will create a layer with the same name with Anno for a suffix in the geodatabase the original layer is stored in. With this layer you can set the display properties in general to display at 1:10000 while your layer itself can be set to 1:24000