I think where many people make an error is only writing to the bigger companies. I don't know much about them but I dare say you would get either the brush off or only working in one area. I'd target the smaller companies. Bearing in mind their budget isn't as big either. You do get exposed to a heck of a lot more though.
Use connections and see what happens. Some smaller companies use the ops job as a training ground/stepping stone to a flying position.