Engine Ventilation Fan -- can it be improved?

[Need_for_speed]

Went for a long drive yesterday and, after parking our Box in the garage, noticed that hot air was still pouring out of the passenger side vent long after the engine was shut down. Which got me to thinking, could the engine vent be improved? Are there any relatively simple improvements we could make to help with thermal management? Some ideas...

It seems, just from a visual inspection, that the tube between the fan and the vent grill is somewhat restrictive. Could it be enlarged?

The grill in the vent covers half of the opening. Could we remove the grill and replace with a screen?

Has anyone hacked the fan so it could be turned on with battery power when the engine is powered off?

Just thinking out loud...