You have to have the lock screen enabled on the S3 for it to stay. You will only have to unlock it once when you 1st put it on. After that it'll stay unlocked. However one problem I have not figured out is that when I switch between my Gear S3 and my Gear S2 my cards disappear on the one I'm switching to. I don't have that problem if I'm switching to back and forth between one of my Gear 2s though. Of course they don't have Samsung Pay on them but when I switch back to the S3 there's no problem.
The lock screen is enabled. The problem is I can't get past the verification screen. I have called about 20 times to verify my card and they tell me my card has been verified yet when I press the back button on the watch my card pops up on the watch but it says verify card.