Yeah, the air will be cooled alot more efficianly, but the point of a larger intercooler is so there will be more space to cool the air from a larger turbo. A larger turbo, has a larger degree of heat, so all that air needs to be cooled down before going into your engine.
If you feel any gains with a larger intercooler, and a stock turbo, then you're not actually feeling a power gain. What you're feeling is the new intercooler doing what the stock intercooler should have been doing in the first place. That's cooling the air efficiantly throughout the power bar. So the power is nice and even, with no drops due to heat soak. Don't get me wrong, a larger intercooler is a very good investment, I'm just saying it wouldn't really show it's worth until you install a larger turbo.