We employ classical molecular dynamics simulations to investigate the molecular-level structure of water during the isothermal compression of hexagonal ice (Ih) and low-density amorphous (LDA) ice at low temperatures. In both cases, the system transforms to high-density amorphous ice (HDA) via a first-order-like phase transition. We employ a sensitive local order metric (LOM) [F. Martelli, Phys. Rev. B 97, 064105 (2018)2469-995010.1103/PhysRevB.97.064105] that can discriminate among different crystalline and noncrystalline ice structures and is based on the positions of the oxygen atoms in the first- and/or second-hydration shell. Our results confirm that LDA and HDA are indeed amorphous, i.e., they lack polydispersed ice domains. Interestingly, HDA contains a small number of domains that are reminiscent of the unit cell of ice IV, although the hydrogen-bond network (HBN) of these domains differs from the HBN of ice IV. The presence of ice-IV-like domains provides some support to the hypothesis that HDA could be the result of a detour on the HBN rearrangement along the Ih-to-ice-IV pressure-induced transformation. Both nonequilibrium LDA-to-HDA and Ih-to-HDA transformations are two-step processes where a small distortion of the HBN first occurs at low pressures and then, a sudden, extensive rearrangement of hydrogen bonds at the corresponding transformation pressure follows. Interestingly, the Ih-to-HDA and LDA-to-HDA transformations occur when LDA and Ih have similar local order, as quantified by the site-averaged LOMs. Since Ih has a perfect tetrahedral HBN while LDA does not, it follows that higher pressures are needed to transform Ih into HDA than that for the conversion of LDA to HDA. In correspondence with both first-order-like phase transitions, the samples are composed of a large HDA cluster that percolates within the Ih/LDA samples. Our results shed light on the debated structural properties of amorphous ices and indicate that the kinetics of the Ih-to-HDA and LDA-to-HDA transformations require an in-depth inspection of the underlying HBN. Such investigation is currently ongoing.