Along with what others said, look under at the wheels where the transmission or differential connect to the wheel. There is a rubber boot, make sure they're not cracked or leaking grease.

Check around the vehicle at the seams of where the body panels and doors meet, the headlights. You're looking for anything that looks like the segments aren't spaced the same. Mainly you're checking for previous damage that may have been repaired. If you see stuff not spaced evenly it could mean it was in a collision at one point. Might not be a problem but could mean damage like the frame is bent which can cause problems.

Smell the inside carpets, maybe if you can make them slightly moist, see if it smells musty or look for rust under the seats where the metal is. Could be flood damage. This can lead to weird electrical problems for example and other problems like odors in humidity.

Pull the carpet up in the trunk, check if it has the jack and spare tire, and check if any water accumulating in there.

EDIT: Also check the ceiling inside along the top, look for stains that could indicate a leaking roof.