Equally unfortunate: there's no substitute for experience. I can't explain how to visualize routing, after 30+ years of sporatic needs, I'm only competent. I still ripup large areas when I see a cleaner pattern to pursue.
I try to work from inner IC pins to the ends. From yours' I see either double sided OR a boat load of jumpers. It's just too busy with individual runs.
Threading between pins works fine when processed by a professional board house. I only try it now, on in-house boards, when one or both pins I'm splitting have no connection (delete its' pad for clearance). Direct photo image transfer and a good etch tank make this better, but I always had a hard time. <<<)))