As far as I know, this is how it works:
You need to move out of line-of-sight (or at least behind some solid cover - I'm not quite sure about that). How much movement you use to do that doesn't matter, as long as it follows any of your movement limitations. Once you are behind cover from whoever you're trying to hide from, you use your Action (or Bonus Action if you have that ability) to Hide. You roll Dex (Stealth) against the target's Passive Perception. If you make it, they can't see you. They may be vaguely aware that you're over there somewhere, but you get whatever benefits (like Advantage on your attack, or whatever) from being hidden. You are spotted if you move back into line-of-sight before attacking.
Outside of combat depends on the mode, I guess. In Exploration mode (which is really used for overland travel, usually) you can use stealth when travelling at a slow pace, which is usually 2/3 whatever speed you'd normally travel at (like 20 miles in a hike instead of 30.) It looks like that goes for city-scale or dungeon scale too. So if you're exploring in 1 minute segments, your average character would go up to 200 feet while sneaking.
That kind of scale is usually used outside of an encounter, though, and usually only used to determine whether you have surprise or not. It also usually assumes that the party sticks together, for the most part. Once you run into something, if you've been stealthy enough (ie beat their PP with your stealth). Then it goes into combat rounds and you have a surprise round first, where you can now move-dash-hide if you're a rogue.
In our specific situation, surprise is probably a given, considering that the goblins had pumpkins on their heads at the point of encounter. The starting point for a rogue character relative to everyone else is more a matter of what everyone was doing at the time the encounter began (was he scouting ahead or sticking with everyone else or looking at something), rather than having anything to do with the rogue's ability to triple-move in combat.
In this case, if the rogue's player thinks they should be further ahead, then they probably should be.
In conclusion: It seems that once an encounter has started, it's more a matter of running from cover-to-cover, rather than sneaking around. In exploration, you sneak around, but that ends once an encounter happens. The grey area between the two is where we've found ourselves, which is why it's a bit tricky.