I'm similarly qualified to you, but a couple of decades down my career. I've worked extensively in GA, bounced briefly off the airline world, but also worked a lot in the military environment.
The jobs people like us can do in the aircraft operational side are relatively out of the core, but nonetheless vital.
In GA, I worked in airworthiness approvals - designing or mostly testing and approving aircraft modifications, or investigating in service airworthiness issues.
In the military environment my role was mostly managing aircraft or equipment testing. A lot of fun it was too, I sometimes still regret leaving that job - although for me personally it was pretty inevitable.
In the airline world people like us are most likely to be in areas such as airworthiness oversight, modification approvals - mods come along all the time in the airline world: for example cockpit changes, trim updates, rectification of design defects that have been found in service.
The biggest problem you'll have as a graduate engineer in GA is that very few organisations are geared up for training graduates - the best bet is to get a job as a deputy or assistant to an experienced professional engineer in something like an airworthiness or design function. Those jobs are like hens teeth - but if you are offered one, grab it, it's a fantastic way into the industry.
In the military environment, there's a much more straightforward graduate entry route at places like Boscombe Down, BAeS, Westlands or Qinetiq. Their websites should show the routes in if you hunt around. (Or there's Engineering Office entry for all three of the services.)
The airlines are probably closer to GA in that there's not much of a tendency to recruit fresh graduates, but there are certainly interesting jobs for good CEng or IEng level Engineers. But there are certainly entry level jobs you can try and shoot for, but you'll need to do a lot of hunting to find those elusive graduate jobs in that environment.
(I started in the military environment with one of Qinetiq's predecessor organisations, before moving later into GA where I was already qualified and capable of practicing independently, and then finally moved into the research environment I work in now. I can't honestly see any other order having worked for me.)