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The cosmic X-ray background (CXB) is dominated by the obscured and unobscured coronal light of active galactic nuclei (AGN). At energies below 10 keV, the CXB can be well explained by models taking into account the known AGN and the observed distribution of their obscuring, line-of-sight column densities, $N_{\rm H,l.o.s}$. However, at energies around the Compton reflection hump ($\sim30$ keV), the models fall short of the data. This suggests the existence of a population of as yet undetected Compton-thick AGN ($N_{\rm H,l.o.s}>1.5\times10^{24}$ cm$^{-2}$) whose X-ray spectra are dominated by the light that has been reprocessed by the obscuring material. In this work, we continue the effort to find and catalog all local ($z<0.05$) Compton-thick (CT) AGN. To this end, we obtained soft X-ray data with Chandra for six local BAT detected sources lacking ROSAT (0.1-2.4 keV) counterparts, indicating potential obscuration. We fit their spectra with Bayesian and least squares methods using two different models, borus02 and UXCLUMPY. We compare the results of the different models and methods and find that the $N_{\rm H,l.o.s}$ is consistently measured in each case. Three of the sources also were observed with XMM-Newton allowing the opportunity to search for variability in soft X-ray flux or $N_{\rm H,l.o.s}$. From this sample, we find one strong CT candidate (NGC 5759) and one weaker CT candidate (CGCG 1822.3+2053). Furthermore, we find tentative evidence of $N_{\rm H,l.o.s}$ variability in 2MASX J17253053-4510279, which has $N_{\rm H,l.o.s}<10^{22}$ cm$^{-2}$.