Investigation of the properties and phase behavior of noncrystalline water is hampered by rapid crystallization in the so-called "no man's land." We here show that it is possible to shrink the no man's land by lifting its low-temperature boundary, i.e., the pressure-dependent crystallization temperature T x (p). In particular, we investigate two types of high-density amorphous ice (HDA) in the pressure range of 0.10-0.50 GPa and show that the commonly studied unannealed state, uHDA, is up to 11 K less stable against crystallization than a pressure-annealed state called eHDA. We interpret this finding based on our previously established microscopic picture of uHDA and eHDA, respectively [M. Seidl et al., Phys. Rev. B 88, 174105 (2013)]. In this picture the glassy uHDA matrix contains ice I h -like nanocrystals, which simply grow upon heating uHDA at pressures 0.20 GPa. By contrast, they experience a polymorphic phase transition followed by subsequent crystal growth at higher pressures. In comparison, upon heating purely glassy eHDA, ice nuclei of a critical size have to form in the first step of crystallization, resulting in a lifted T x (p). Accordingly, utilizing eHDA enables the study of amorphous ice at significantly higher temperatures at which we regard it to be in the ultraviscous liquid state. This will boost experiments aiming at investigating the proposed liquid-liquid phase transition.