Let $U_q'(\mathfrak{ g})$ be a quantum affine algebra of untwisted affine $ADE$ type, and $\mathcal{ C}_{ \mathfrak{ g}}^0$ the Hernandez-Leclerc category of finite-dimensional $U_q'(\mathfrak{ g})$-modules. For a suitable infinite sequence $\widehat{ w}_0= \cdots s_{ i_{ -1}}s_{ i_0}s_{ i_1} \cdots$ of simple reflections, we introduce subcategories $\mathcal{ C}_{ \mathfrak{ g}}^{ [a,b]}$ of $\mathcal{ C}_{ \mathfrak{ g}}^0$ for all $a \le b \in \mathbb{ Z}\sqcup\{ \pm \infty \}$. Associated with a certain chain $\mathfrak{ C}$ of intervals in $[a,b]$, we construct a real simple commuting family $M(\mathfrak{ C})$ in $\mathcal{ C}_{ \mathfrak{ g}}^{ [a,b]}$, which consists of Kirillov-Reshetikhin modules. The category $\mathcal{ C}_{ \mathfrak{ g}}^{ [a,b]}$ provides a monoidal categorification of the cluster algebra $K(\mathcal{ C}_{ \mathfrak{ g}}^{ [a,b]})$, whose set of initial cluster variables is $[M(\mathfrak{ C})]$. In particular, this result gives an affirmative answer to the monoidal categorification conjecture on $\mathcal{ C}_{ \mathfrak{ g}}^-$ by Hernandez-Leclerc since it is $\mathcal{ C}_{ \mathfrak{ g}}^{ [-\infty,0]}$, and is also applicable to $\mathcal{ C}_{ \mathfrak{ g}}^0$ since it is $\mathcal{ C}_{ \mathfrak{ g}}^{ [-\infty,\infty]}$.
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