Well, we sure know that the Greeks put the scale onto the linothorax. Here's a nice vase painting to illustrate their use:



There are several things to note about the man on the left:

1) The detached shoulder piece. It may just be the coloring options given to Sosias, but we have an interesting color for the material on the underside of this armor--leather perhaps? The color of the hardened inside of the linothorax, before having the outside layer bleached? Whatever it is, we know that there are scales implanted on the outside of it.

2) The stomach is also covered in scales, and, based on the patterning above and below the scales, would seem to be a linothorax base. We can't say for certain. It could just as easily be leather, or a simpler cloth used mainly as a base for the scale. But it looks like a linothorax base. However, we can say that the underside of the connector piece for the shoulder (see other shoulder piece for reference) is a different color than the underside of the main piece, which is a lighter color. Leather vs. linen? Hardened linen vs. linen?

3) Note that below the second set of designs we see scale pteryges. There are two interpretations I can think of: a) it is a separate piece attached around the waist, b) the scale we see on the stomach continues here, it is more like a corselet attached to the chest, bound by the design we see around the waist, which is a belt, and not the bottom of the linothorax. Now, even with option a) its still probably attached to the thorax itself, but its unclear whether the base for these scales would be linen or leather, both were used in pteryges, so either is possible.

4) the chest section. we can see where it ends, how it extents to the base of the shoulder, at which point it relies on the shoulder pieces to manage the transition from front to back. We can see a separate border, or hem, going around the edge, with some sort of light (even filmy) garment underneath. This chest section itself may feature some form of "quilting," or rather, brick-patterned stitchwork, which I find more likely--considering the depiction--than merely designs drawn on the linothorax: if designs they do not fit with the theme of the things on the cuirass which are clearly just designs, and are considerably more crowded than anything else of which I am aware.

You'll also notice the exclusively scale construction of the other thorax, I'm not really sure what all is going on there, mainly because the perspective is a little problematic.