  A major question in the study of black hole binaries involves our
understanding of the accretion geometry when the sources are in the hard state.
In this state, the X-ray energy spectrum is dominated by a hard power-law
component and radio observations indicate the presence of a steady and powerful
compact jet. Although the common hard state picture is that the accretion disk
is truncated, perhaps at hundreds of gravitational radii from the black hole,
recent results for the recurrent transient GX 339-4 by Miller and co-workers
show evidence for optically thick material very close to the black hole's
innermost stable circular orbit. That work focused on an observation of GX
339-4 at a luminosity of about 5% of the Eddington limit and used parameters
from a relativistic reflection model and the presence of a soft, thermal
component as diagnostics. In this work, we use similar diagnostics, but extend
the study to lower luminosities (2.3% and 0.8% Ledd) using Swift and RXTE
observations of GX 339-4. We detect a thermal component with an inner disk
temperature of ~0.2 keV at 2.3% Ledd. We detect broad features due to iron
Kalpha that are likely related to reflection of hard X-rays off the optically
thick material. If these features are broadened by relativistic effects, they
indicate that optically thick material resides within 10 Rg down to 0.8% Ledd,
and the measurements are consistent with the inner radius of the disk remaining
at ~4 Rg down to this level. However, we also discuss an alternative model for
the broadening, and we note that the evolution of the thermal component is not
entirely consistent with the constant inner radius interpretation. Finally, we
discuss the results in terms of recent theoretical work on the possibility that
material may condense to maintain an inner optically thick disk.
