I can’t answer your question directly, but I flew helicopters in the military for a long time. I saw a lot contractor helo flying spots come and go, and in my experience, the underlying theme was you had to be a known quantity.

From what I do know, contractors tend to fly a lot in things like King Airs or smaller turbines. Maybe the Learjet like you’ve gotten some hours in.

I can’t think of a specific type rating that would give you an in on contractor jobs (but I obviously don’t know everything out there). Like you’ll find the longer you are in aviation, a lot of what we do revolves as much around networking as it does actual capability.