We continue the study of extended T-systems of quantum affine algebras. We find a sub-system of the extended T-system of the quantum affine algebra $$U_q \hat{\mathfrak {g}}$$ U q g ^ of type $$C_3$$ C 3 . The sub-system is consisting of four systems which are denoted by I, II, III, and IV. Each of the systems I, II, III, IV is closed. The systems I–IV can be used to compute the $$q$$ q -characters of minimal affinizations with weights of the form $$\lambda _1 \omega _1 + \lambda _2 \omega _2 + \lambda _3 \omega _3$$ λ 1 ω 1 + λ 2 ω 2 + λ 3 ω 3 , where at least one of $$\lambda _1$$ λ 1 , $$\lambda _2$$ λ 2 , $$\lambda _3$$ λ 3 is zero. Using the systems I–IV, we compute the characters of the restrictions of the minimal affinizations in the systems to $$ U_q \mathfrak {g} $$ U q g and obtain some conjectural decomposition formulas for the restrictions of some minimal affinizations.