I installed the Fantastic Ultra Breeze hood on my Fantastic Vent Fan, Its easy to remove and is large allowing lots of air flow and angled to keep the rain out.
Your cheapie bathroom vent fan is probably in a standard 14x14 hole and you could install a better fan if you desired.
MaxxAir makes one that the hood is actually part of the unit and raises up with it. It does need several inches clearance behind the fan, so no vents, antennas or whatever within about 6 or 8 inches of the rear of the vent hole, but that is usually not a problem.
https://www.airxcel.com/maxxair/products/fans/maxxfan-deluxeScroll down to the series of pics that show it both open and closed.
Charles