No theater (that I know of) has planted its flag in that uncharted territory, but it may be the final yet-unexplored theme/gimmick for the next generation of whippersnappers who want to start a company: "World premieres are great and all, but we only do second productions."
In recent years, at least in Seattle, playwrights have made headway in convincing theaters to produce more world premieres, just as actors have made serious headway in convincing them to cast locally. (Compare the season of almost any regional theater these days to an average season in the late 1990s and you'll see the difference.) Some of that is about economics, of course—the local yokels are cheaper to hire than flying in Dame Thespian and putting her up in an apartment for a few months.
But in the past few years, I've begun to hear playwrights grumbling about the lack of opportunity for second productions.
There's a discussion happening about that on Facebook right now—but I shouldn't tell you where or who or what they're saying because the last time I did that, people went bananas and excoriated me for violating something called "Facebook privacy"—but the kickoff post, by a newish playwright, says it all:
... already I'm learning that having had a production is more of a liability than a help. It seems even the tiniest theatres are very concerned with claiming premieres.
Which makes sense—theaters are comfortable with known quantities because audiences are comfortable with them. Some theatergoers are categorically in the camp of "I saw a new play once and it was boring, so new plays are boring." Like this guy, responding to my "produce new plays" suggestion in the Ten Things Theaters Need to Do article (that has gone strangely viral all these years later):
And theaters, in part due to economic pressure and in part due to community pressure, have developed a new interest in world premieres. But that interest can tip over into fetish territory stranding the in-between plays—second productions, third productions—in uncomfortably becalmed seas.
Of course some theaters—such as Woolly Mammoth in D.C.—pride themselves on doing world premieres and second productions, but a scrappy new company might consider making that its entire reason for being.
There are some things to recommend that approach:
One, it narrows down the field that your literary managers and artistic directors have to keep track of (I don't know if anyone's done a survey, but I'm guessing the number of unproduced plays floating around profoundly dwarfs the number of once-produced plays).
Two, if a play has been produced, you know it's at least produceable and that it had enough meat to convince some group of people to take a crack at it.
Three, you have the benefit of their experiment. You can look at old reviews and call up the playwright, directors, and designers to ask what worked, what didn't, and what they'd do differently if they had it to do over again—the magical sweet spot this fictional new company would be sitting in.
Fourth, you'd be a novelty (which is good for getting attention) and you'd be beloved by the living playwrights of the world (as long as you don't screw any of 'em).
A young new company looking for a hook could do a lot worse than "we only do second productions."