Can a Major Image illusion move outside of the range in which it was originally cast?


Major Image (relevant sections):

You create the image of an object, a creature, or some other visible phenomenon that is no larger than a 20.foot cube. The image appears at a spot that you can see within range and lasts for the duration.

As long as you are within range of the illusion, you can use your action to cause the image to move to any other spot within range. As the image changes location, you can alter its appearance so that its movements appear natural for the image. For example, if you create an image of a creature and move it, you can alter the image so that it appears to be walking.

At Higher Levels: When you cast this spell using a spell slot of 6th level or higher, the spell lasts until dispelled, without requiring your concentration.

I initially presumed this to mean you can create an illusion limited by the 20ft cube maximum size, and then the illusion can move within the original 120ft casting radius, with no option to leave it. But upon reading it again, it seems that if the caster is within range of the illusion, the character can command the illusion to move outside any initial range, and indeed can have the illusion follow the character around the map for miles. Is this correct?