I would think very deeply before making that move to the GA. Been there myself for 9 years. I must admit, i did not fly for a well known outfit with big shiny privet jets and was flying a small jet. The good thing is you see a lot. We were flying to places that you would normally never visit. But like everything, novelty wears off after a while and you start thinking about other things that are getting more important to you. Lifestyle, roster, money, social life and yes, having hobbies without being disturbed.
Afterwards i joined the airline. First pax airline, then cargo airline. BEST CHOICE EVER!!!!!! I must admit: it is also not all that perfect and sometimes tedious. Yes we do fly all the time the same routes and yes, sometimes i miss those GA flights to these odd places. But the quality of life i have now: much better and much more stability.
I would say it is for an outsider difficult to get in GA and flying immediate a Gulfstream or Global express. The best GA jobs are simply not advertised and it is all about connections. You will high likely start at the bottom, like a citation or king air (which can be also fun!!!). You will get to see the bits and pieces and everything that involves GA. And you have to be extreme flexible and planning your private life is almost impossible. As a younster who is fresh in the job this is not hard, but for an old timer like me......
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Most of the GA outfits are just crap. You simply dont want to work for them. Low salary and they push you all the time. In the end, it is just work. Nothing more. Perhaps 1 exception: Netjets. In general i hear a lot of goods stuff about them. But i must warn you. Even though Netjets is a great place, if you ever get made redundant, you are left behind with a useless typerating. So make sure you keep yourself current on that Airbus/Boeing.
My best advice is to stay with your airline gig. Try to go parttime and explore GA as an freelancer. Betting on 2 horses never hurts.