In dry climates it's common for people to use evaporative ("swamp") coolers; same idea as sweat evaporating off your skin cooling you off. The downside is that they increase humidity in the air, and if it's already pretty humid then that can make things uncomfortable. This means evaporative coolers are rarely used outside arid regions.

Whole-house fans use a completely different principle: instead of cooling air they just trade it. If it's 85F at the day's high and 65F at the low, running a lot of cool night-time air through the house can cool the structure down enough that it stays cool all day. The main limitation is you can't even in theory cool the structure below the day's low, and in practice you're going to be several degrees above that.

But what if we combined these? Running an evaporative cooler during the part of the day when the house is closed up wouldn't work well, but if you ran one at night all the extra humidity would just get blown out through the whole-house fan. This seems like it should let you cool your structure to below the night-time low.

I'm not sure exactly what the mechanism for this would look like. For example, consider a very simple house:

   +-------+--------+
   |    ---+---     |
                   \ /
  -->            --( )--
                   / \
   |                |
   |                |
   +----------------+

We have an open window on the left side to let air in, a ceiling fan to mix the air around, and a fan on the right side to blow the air out again. Where should the damp evaporative surface be?

One option would be cooling the air, with something like a hanging towel. I think this would work, but would need a lot of water for the amount it cooled the structure: you're cooling the air, and most of the air is being immediately blown out again.

Another option would be to use the wall, ceiling, or floor as an evaporative surface. The advantage here is that you should have better heat transfer, where you draw more heat out of the structure than doing it indirectly via the air. Maybe make the walls out of something that doesn't mind getting damp (ordinary paint?), and gently mist them all night long at whatever rate balances evaporation? You'd want to stop a while before turning off the whole house fan, to reduce humidity levels down to something pleasant. You probably also wouldn't want to do this in sleeping spaces.

Would this work?