Married or Single? Willing to travel? Ambitious? Confident?
You may be approaching this all the wrong way. Most companies are run by donkeys who hire donkeys, the pay is crap and the smart money goes to a select few... Think about that for a moment... Finding work is intimidating, but once you have a job, how quickly do you outgrow your job? Be honest with yourself about that..
If you're better than most of the other techs around you after 6 months, then stop underselling yourself even if you are a generalist! It doesn't matter anyway, the crucial question is this: Are you working in an important area or an important project? If not, then why not?
Your work hunt attitude comes across as slightly vulnerable, so for starters be mindful of this and purge any quiet defensiveness and don't give off the vibes of the guy whose desperate for a date. Confidence is everything! For starters stop apologizing like someone singing badly at karaoke and tell us what your goals are, what your dreams are, and keep it practical by way of including a salary / contract range and desired positions, for now and in 5 years etc?
You sound conscientious, the type of guy that will deliver on crucial projects, so try to leverage that. I would actively seek out recruiters struggling to find staff for distressed projects, but only accept those with attractive payouts and seniority. Once you succeed at this a couple of times, no one will ever question your weak points again and they'll see the battle scars. Why? Because when you talk about succeeding in difficult scenarios you take on a whole other persona that is quite powerful in selling yourself without actually selling yourself or selling out etc. So come across as person who delivers. Full stop! ... Because that's what this IT game is about really.
IT is such a shit-hole career now. On a personal note I've stayed in contracting most of my life and tried to avoid those gigs that might be a nest of vipers. Instead I set my sights on tax-free havens such as HK, Dubai, Bermuda. Sometimes I won these, sometimes I didn't like anything else in life. But I kept my high standards regardless. To survive I needed a decent float, at least one or two lucrative earners every couple of years where I could bank it all. Keeping the float intact was surprisingly easy as I was just too busy to blow it all... But here's an Important caveat. I only managed to do this because I wasn't married and was mostly unattached during high income years... Otherwise I've have gotten more than a few P45's in my relationship Christmas stocking!
One last option that some seek out or like me accidentally fell into, is to work for trading shops. But this business is changing so fast its hard to know where the hot areas are right now. After the meltdown hedge funds, LBOs and M&A and HFT shops were the places to go for techs, but I'm not sure anymore. Bloomberg write articles about this from time to time so look under 'view'. But be wary accepting gigs at any of the big banks. If you work there you probably won't even make it to the middle or back office never mind the front desk. Overall, its not very nice work either, but early retirement is definitely on the cards... Strong math is a mega-advantage, but the ability to communicate highly technical ideas to distracted traders in a fast, fun and humorous way gets you noticed!
But mostly, good luck!