Given the heart of your question is to "Expand the analog signal in hardware", then what you really want is to increase the ISO setting. At ISO 100, a camera will record the image signal using the maximum available range. In a scene like you described, one that is naturally low in contrast, if your total sensor dynamic range is around 11 bits of data, when the scene only contains 4 bits, you'll end up with most of that information bunched up in one area or the other.
You effectively have a dynamic range problem, although one a bit different than what most photographers encounter: You have too much. For each stop of increase to ISO, dynamic range drops by approximately a factor of two. If you are using 1/2 of your available dynamic range at ISO 100, increasing ISO to 200 would reduce the useful dynamic range by a factor of two, in line with what you need. Similarly, if you are using 1/4 of your available dynamic range at ISO 100, increasing ISO to 400 would reduce the useful dynamic range by a factor of four, again in line with what you need.
This is because the camera changes the white point, or "saturation level", when you change ISO. Changing ISO does not increase the sensitivity of the camera, it is simply a change of instruction to the analog amplifier. If your camera's maximum per-pixel charge level (FWC, full well capacity) is 60,000 electrons, and a "saturated" charge of 60,000 electrons gets converted into a digital unit of 2^14 (16,384), increasing to ISO 200 means that the maximum allowed charge per pixel is 30,000. Now a "saturated" charge of 30,000 electrons is converted into DU 16,384. Similarly, increasing ISO to 400 means that a "saturated" charge of 15,000 electrons is converted into DU 16,384.
When you increase ISO, you have to adjust your exposure to compensate. You will have to "fit" the exposure into the available dynamic range. Increase ISO high enough, and your blacks will ultimately reach the noise floor, and your whites will begin to clip...you've effectively used your entire dynamic range, all in-camera, changing nothing but the analog amplification. It should be noted that at very high ISO settings, usually above ISO 1600 (although newer Canon cameras should be good up to ISO 6400) increasing ISO further results in a mix of per-pixel amp, downstream amp, and possibly even digital amp simultaneously.