Omer, I'm not sure what you mean about cooling to ambient.
What can you can cool the first stage gas to is dependent on your cooling medium and how close of an approach you want to design your cooler for be it air, water or some other medium. The closer an approach, the more area you need but you save on compressor Hp, it's a trade-off.
For air coolers, 10C approach is a common lower limit, 20C would not seem unrealistic. If someone is trying to tell you they need the interstage gas cooled to ambient air temperatures using an air cooler (and I'm guessing this from your post), they don't know what they are talking about. It's impossible.