There's no official term for this. I also think of it as "progressed
improper".
Rick Mohr's written a number of dances like this -- for instance:
Leave the Wine, My Marbles Have Many Moods, Synchronicity. These are
all dances that can start proper, but after the first time default to
progressed improper. (Another curiosity that starts improper and then
converts to other formations is "People of the Pond" by Jim Saxe.)
It's an interesting choice (all of Rick Mohr's dances could be
Becketed), but I don't see problems on the dance floor when they're
done, as long as dancers are reassured it's not a bug.
More and more, I think not about what formation the dance is in, but
what formation the dancers progress with -- whether they progress/get
back into the dance improper, indecent, becket, or something else.
-Chris Page
San Diego
p.s. The square through dance from an earlier post is a probably
mistranscribed version of "Stir Crazy" by Erik Weberg. The original is
Becket.