I'm not a "sound" guy so probably not the best person to make suggestions, but I can share what I observed when I worked with Angelo Panetta on SO. Angelo knows the whole sound design thing inside and out.
Imagine what kind of environment your scene sits in and then imagine what sounds would be in that environment. Environment sounds don't have to be loud or very noticeable. Even barely audible sounds will have an impact when they are all layered together.
Some common elements for an outdoor scene might be, wind blowing, leaves rustling, sounds from birds and other critters, water running in a stream etc.
An inside environment might include things like a clock ticking, a radio or TV faintly audible, a refrigerator, a fan or air conditioner, cars driving by outside etc.
A large hallway might include echoes from voices or footsteps far down the hall, doors opening and closing, elevator sounds, muzak playing over a speaker system etc.
Also when a character moves they make sound. Footsteps, the sound of clothing brushing against itself, the sound of a person rubbing his/her face or scratching his/her head. These types of sound, from my limited experience, are often exaggerated.
Is that the kind of thing you were looking for?