Supermarkets are hopeless for this sort of thing. We have a Tesco Extra the size of a small village near to where we live, but half the time it doesn't even have lettuce, and when it does, it's often iceberg. Pak choi? Endive? Forget it!
You need to go to a big greengrocer if you can find one near you, the organic faffy hippy types are good for variety. I don't know where you guys are but if you happen to be anywhere near Glasgow, Roots & Fruits on Great Western Road is great for a variety of veggies; so is the Peckham's greengrocer on Hyndland Road. Or try a fruit market. We have one of those not too far from where we live, I've never visited it but I intend to. There are probably fruit markets in most cities. Go somewhere, anywhere but a supermarket, and you'll be surprised how much better the taste is, and how much wider the range is.
Sorry I can't tell you the names of any chains which sell good veggies, because that's about the size of it. Supermarket chains don't sell good veggies.