There are several options to increase the signal size:
1) increase the length of the path (look at the slab from side to side instead of from the front - you may have seen a "green edge" on a glass coffee table - that's because you see light coming through a much thicker slab of glass)
2) Use multiple sheets of glass on top of each other. Be careful about multiple surface reflections; you might be able to angle your laser at the Brewster angle to eliminate surface reflectivity
3) Put the sheet in a Fabry-Perot type setup where you have multiple reflections between the mirrors (and thus, multiple passes through the glass). As before, you have to worry about the surface reflectivity of the glass.
In all cases you don't change the absorbtion per unit length, but you increase the length.
The other thing you can do, and the thing I would prefer, is measure the difference between the incident and transmitted laser. To do this, you have to expand a fraction of the incident beam, and make it interfere with the expanded transmitted beam. First you adjust the relative intensities so the fringes have a "perfect black" when you have no glass in the way; then you add the glass, and now the intensities of the two beams will differ by a small amount, and this will be the new intensity of the minimum in the interference pattern. Measuring the maximum and minimum intensity will give you (with some simple math) the fraction of the absorbed intensity.
This latter approach has an enormous advantage in that it measures the difference directly. That is usually the best approach in any accurate physics experiment. I like to use the example of measuring the thickness of a piece of wall paper. You could measure the distance from the opposite wall to the front of the paper, and to the wall behind the paper, and calculate the difference. Or you could measure the sheet of paper directly with calipers. No guessing which will work better.