We observed Circinus X-1 twice during a newly reached low-flux phase near zero orbital phase using the High-Energy Transmission Grating Spectrometer (HETGS) onboard Chandra. In both observations the source did not show the P Cygni lines we observed during the high-flux phases of the source in 2000 and 2001. During the prezero phase the source did not exhibit significant variability but did exhibit an emission-line spectrum rich in H- and He-like lines from high-Z elements such as Si, S, Ar, and Ca. The light curve in the postdip observation showed quiescent and flaring episodes. Only in these flaring episodes was the source luminosity significantly higher than observed during the prezero phase. We analyzed all high-resolution X-ray spectra by fitting photoionization and absorption models from the most recent version of the XSTAR code. The prezero-phase spectrum could be fully modeled with a very hot photoionized plasma with an ionization parameter of log ξ = 3.0, down from log ξ = 4.0 in the high-flux state. The ionization balances we measure from the spectra during the postzero-phase episodes are significantly different. Both episodes feature absorbers with variable high columns, ionization parameters, and luminosity. While cold absorption remains at levels quite similar to that observed in previous years, the new observations show unprecedented levels of variable warm absorption. The line emissivities also indicate that the observed low source luminosity is inconsistent with a static hot accretion disk corona (ADC), an effect that seems common to other near-edge-on ADC sources as well. We conclude that unless there exists some means of coronal heating other than X-rays, the true source luminosity is likely much higher, and we observe obscuration in analogy to the extragalactic Seyfert 2 sources. We discuss possible consequences and relate cold, lukewarm, warm, and hot absorbers to dynamic accretion scenarios.