A 1.4x teleconverter would work, but you lose out on IQ. I had had a 2x Tamron that I used on my 70-210/4. And it would get to 420mm, but it was pretty poor IQ. The magic number for Sony's to retain AF is f/8 - anything beyond that, and the bodies will simply not allow focus. To be honest though, if I were going to do it, I'd do it right and get the Sony branded (and supported) ~$500 T/C. The off-brands, aside from the Kenko Pro, just don't work all that well.
However, if I need to really reach out, I can use a virtual T/C built into my body, but then I have to shoot jpegs. Its a pretty cool little feature when I need the extra reach, even if the resolution drops. For 1.4x, I would be dropped down to 12mp, for 2x I would be at 6mp. For those extra reach moments, I don't know if I would go down to 6mp, but I may for 12mp. Getting a shot framed right at 560mm is awfully appealing, jpeg or 12mp notwithstanding.
Minolta 50/1.4, 17-35/2.8-4, 24-105/3.5-4.5, 35-105/3.5-4.5, 80-200 HS APO G, 100-300 APO
Tamron Di 90mm/2.8, 28-75/2.8