According to the Kingsport Horror FAQ, you get an explored token anytime you move from another world to a location with a portal to that other world.
For example, if an arkham encounter sends you to the dreamlands and instructs you to return, and the location you're in has a dreamlands portal, you would count as having explored it. Another example is one Abyss encounter which instructs you to move to the black cave; if they black cave had a portal to the abyss, you get an explored token when you move there.
The gate box is an obvious exception to the rule that the portal must match the outer world you just came from. Example, if you enter a dreamlands portal, move through the dreamlands, and exit through a R'lyeh portal using the gate box, you get an explored token and can attempt to close.
But to what extend would the gate box help? If you get the arkham encounter that sends you to the dreamlands and back, and you land on a portal to City of the Great Race when you come back, do you get an explored token if you have a gate box? If you get the abyss encounter that sends you to the black cave, and the black cave has a non-abyss gate, do you get an explored token if you have a gate box?