  We study the spectrum of forcing notions between the iterations of
$\sigma$-closed followed by ccc forcings and the proper forcings. This includes
the hierarchy of $\alpha$-proper forcings for indecomposable countable ordinals
as well as the Axiom A forcings. We focus on the bounded forcing axioms for the
hierarchy of $\alpha$-proper forcings and connect them to a hierarchy of weak
club guessing principles. We show that they are, in a sense, dual to each
other. In particular, these weak club guessing principles separate the bounded
forcing axioms for distinct countable indecomposable ordinals. In the study of
forcings completely embeddable into an iteration of $\sigma$-closed followed by
ccc forcing, we present an equivalent characterization of this class in terms
of Baumgartner's Axiom A. This resolves a well-known conjecture of Baumgartner
from the 1980's.
