Guess – Enhance Athmospheric Water Harvesting In A Volume Of Higher Air Pressure?

I’ve seen small air to water harvesters using peltier elements. Here they let the water in the air condense on the cold side. Since a bicycle pump gets warm when you compress the air. I wonder. Would you not get a higher efficiency if you tried to cool this warmer air. Also since the volume is smaller maybe its easier to extract water from it once its colder.

So the idea is to have a air compression water extraction cycle. Air is compressed to a container then water is extracted then the air is let out. After that the cycle repeats with new fresh compressed air coming in.

Looking at some peltier cooling experiments I noticed that the thickness side of the ceramic peltier plate was hot. Maybe there is some efficiency gains one can do if this heat is diverted away from this small but hot side.