I never played TNE, but I would guess TNE because that's a prime motivation for adventuring with that edition. Whole sectors have been cut off from each other. Many of the game goals in adventures were to reconnect.
I don't really think it matter what edition of the rules you use, but there are probably different eras in the official timeline that are more about exploration. I mean, it's a big galaxy. Scouts came out as a book right in the smack dab middle of the Traveller Golden Era. The Zhodani were known for their galactic core probes. Just go to the edge of explored space, and start exploring.
As for eras, obviously Milieu 0 (where T4 was set). Or, just after the Virus (TNE, as I just suggested).
Really, any era would work. Pick your favorite rule set, add in your favorite era, then find the best place to do some scouting. In no era is the entire galaxy explored (or even close to being explored).