Beany Has a Secret Life by Lenora Mattingly Weber (1955)
and it revolves around a very "hush-hush" new club. When invited to become one of the privileged few members, Beany is enthralled--life had been looking very down, with her favorite date going to college halfway across the country, and her father, newspaperman Marty Malone, springing a new stepmother on the family, to mention but two of Beany's current problems.
However, before many weeks have passed, the club has added to the complications in Beany's life; she's quarreled with Adair, her pretty stepmother; and, in fact, things have gone from Bad to Worse to Dreadful.
But to quote one enthusiastic reviewer, the untangling of this very snarled situation "makes a thoroughly absorbing story...Beany fans are in for a treat!" (from the back cover)