I never wear a white coat these days (not appropriate to my speciality at all) but even when I did internal medicine, I generally avoided them because they were hot and uncomfortable and generally fit badly!
Frankly I would think a pocket square in the lab coat pocket would look laughable! However, if you're wearing suit jacket underneath, you could have one there (I never did, I always took any jacket off before donning the white coat). You can make things interesting with tie clips, cufflinks, etc instead, which is what I used to do. I quite like the idea of a red carnation in a white coat, though I don't think I'd have the testicular fortitude to try it! Maybe if/when you're VERY senior!
The only accoutrement a white coat needs is a stethoscope hanging nonchalantly round the neck. Gold-plating a la Romano from ER entirely optional!
Though one of the cardiothoracic surgeons at an old workplace of mine DID have one such stethoscope! I must admit, I loved the bling!
(of course you'd only be able to wear the gold-plated scope when your watch and belt buckle are also gold... otherwise you have to stick to the regular steel...