Ive just checked both of my psu fans and one of them is blowing out and the other Ive found not to be working!
The air in the case will be cooler than the air in the psu. I personally would add a fan taking air out the case into the psu with the original psu fan exhausting it out the back.
Looks like I got beat to it, but its made me think about what you mean by bottom. I thought you meant bottom of the psu but it seems you mean bottom of the case.
On the motherboard, there is usually a small connector socket for fans, labelled as SFAN or similar. If it is free, then you can plug the fan into that. If not then some fans have a connector which allows them to tap into the 5v rail from the power supply. (molex connector)