I'm 5'-8 3/4" and I usually ride a 56cm as a max (roomy) size, but I can ride down to 53cm with no problem, as long as the stem height (with longer forward extension) and seatpost/saddle are adjusted correctly.
Usually, most people have a range of sizes they can work with as long as the compnents can be adjusted to compensate as you go up or down a size or two.

Chombi

You need a fitting, not advice from the internet. And if you can afford it buy new and get the warranty, used bikes have no warranty.

My Moto B is 58 and while I rode it for years, it's too tall for my 32" inseam but I've long arms, long feet (13AAA) and torso (I'm 5'10") so it worked. My 'Nago is 56 and fits much better though it sure feels smaller.

54.

For me, it's easier to get comfortable on a frame that's one size smaller, than one size larger. Just raise the saddle and get a longer stem. Also, the smaller frame will weigh less and probably be a little stiffer.

Cycling inseam measure floor to pubic bone. Pant inseam tends to be 1.5-2" shorter than cycling inseam. A 32" pant inseam makes you situation more difficult since it means you have really long legs for your height. I don't think you'll be able to fit a 56, the reach will be too long. So I think you may have to get a relaxed geometry 54 and run a riser stem to get the seat level with the bars. Don't even consider a 54 with less than a 160mm tall head tube.

6'3'' on a 56cm... bike fit is more important than the marked size.

Thanks, great info. My LBS has a red 54 Synapse 7 which really knocks my socks off, but after riding the 105 components on my SL1, the Sora stuff doesn't shift nearly as well. If the red bike had 105 or Ultegra stuff, it might be here already....

Yeah, Sora is hard to justify buying if you can get anything higher.