The funny thing is, Season 1 definitely had this feel - despite featuring plots that explicitly took place well within already explored space!
After "Encounter at Farpoint", which itself takes place at an already known world, only "Where No One" and, somewhat surprisingly, "Symbiosis", feature locations previously truly unknown to the Federation or substantially distant of well-established UFP strongholds. "Justice", while featuring an all-new world, takes place right next to a UFP colonization project; "When the Bough Breaks" describes the discovery of a mythical planet a few hours
away from the beaten path; "The Arsenal of Freedom" involves a "mystery" world previously visited both by long range probes and a starship and well known by reputation; "Skin of Evil" finds our heroes within shuttlecraft range of some conference location or another...
It's actually rather amusing how little is made of the fact that the two worlds of "Symbiosis" are new to the Federation.