bravobravo,
ANY additional experience to your initial training is a good thing. Why on earth could someone possibly think it's not? There's a FI sponsorship in the UK, did you know? Try! You might get lucky.
An another note, you're in your early 20s, young man, supposedly you managed to secure yourself that 737/320 job now. What then? You will have - assuming you keep your medical - 35 years of flying ahead of you. You will make it to captain within say ten years, what then? 25 years of doing the same thing day in day out? All novelty of flying a new aircraft will have worn off after I would say 1-2 years. Nothing new then. Day in, day out, same sh**.
Progressing up the latter is a damn good thing. Teaches you A LOT. And: you will appreciate all the goodies your colleagues take for granted and chances are you'll be one of the few happy pilotos in the company.
Also, climbing the ladder will earn you respect, as people know how hard your way up there has been!
I myself am flying a light twin. You know what, you're going to laugh but even if offered a jet job now, I would not want to take it. I really enjoy the type of flying I do and I am learning A LOT. Next step has to be a TP, do that a couple years, maybe gain a command, who knows. Bottom line is, it's exciting, always something to look forward to, to work towards.
And on a last note, a couple years ago, I was offered a job on another light twin. My first contact was a mere phone call to the company and I happened to get the owner on the phone right away. I had about 400h TT by then, most accrued on SE work (my first flying job was SE but no FI work). I was asked to submit my CV then invited for interview a few weeks later. They were not really looking but the guy was interested I could tell. A months later another interview, going up with him showing him that I could maintain a track, etc. Another six months later and I was finally offered the job. So it took me nearly a year to get that offer. But you know what? Did I take it? No. Got some other offer and went for that...
All the best mate! You'll be fine. Just keep flying, keep flying! If I were you, don't just fork out the money for a FI rating. Go join a gliding club first and go flying and meet people and try to find a flying job without spending an extra 7k. Many GA / airline pilots are also glider / ultra light pilots. Go find and meet them. That's the only way. But doable.
