I wouldn't make it a question of timing as much as a question of who you want to do your surgery.
John Davis did mine at M. D. Anderson. I think he is an excellent surgeon and a nice guy.
However, I have had many hassles with his support staff (which may be more of a reflection on me than on them, of course).
Also, my experience with M. D. Anderson Hospital was not good at all. I don't have time to tell you all the stories. I'm painting with a very broad brush here, and there are certainly some fine and dedicated people working there, but M. D. Anderson is a large state bureaucracy like the Department of Motor Vehicles. They are the best in the world at what they do (which is good) but they know it (which sometimes isn't good). But again, my experience may not have been typical, so get some other opinions.
I would say that if someone has advanced cancer or a rare type of cancer, then by all means go to MDA and just resolve to put up with all the foolishness. But for a more "routine" cancer - comparatively speaking, of course - I would at least look into Memorial Herrmann in Houston, which has put many millions of dollars into its cancer programs and staff in recent years and is making a concerted run at MDA's premier status around town. I have heard that they accordingly make a point to be very user-friendly. I've also heard good things about Dr. Brian Miles at St. Luke's or Methodist (I forget which).
But again, the bottom line is that I think John Davis at MDA is very good. And in the end it's your surgeon who really counts.
I have no experience with Dr. Fagin in Austin.