In 3D concrete printing, fast structuration is a prerequisite for ideal buildability. This paper aims to study the impact of inorganic additives, i.e., CaCl2 and gypsum, on structural build-up and very early-age hydration of limestone-calcined clay-cement (LC3) pastes within the first 70–80 min. Results show that, increasing the dosage of CaCl2 or gypsum can accelerate storage modulus G' and static yield stress evolution with time, as well as increase chemically bound water (H) content and total specific surface area (SSAtotal). Furthermore, good correlations were found between G' and H content, as well as static yield stress and the ratio of free water content to SSAtotal. The acceleration by CaCl2 can be attributed to stimulating C3S and C3A hydration and promoting crystal formation, i.e., ettringite, portlandite, and Friedel’s salt. Additionally, the increase in gypsum percentage led to a large amount of unreacted gypsum in the system, resulting in an increase in SSAtotal.