For box dimensions (upper and lower), we consider the classical intermediate value question of the dimension theory: is it true that in a metric compact space X of the box dimension α (upper or lower) for any non-negative number β not exceeding α, there exists a closed subset F whose corresponding box dimension is equal to β?
For the upper box dimension, a positive answer to this question was obtained in the joint work of the author and O. V. Fomkina. However, this statement is not true in the general case for the lower box dimension. Moreover, it is known that in a wide class of metric compact spaces there exist closed subsets with lower box dimensions of all intermediate values. In this paper, a sufficient condition is obtained that ensures that a fixed number r belongs to the scale of intermediate values of the lower box dimension. Namely, it is proved that if in X there exists a closed subset F whose upper box dimension is less than r and the lower box dimension of any closed ε-ball F is greater than r, then there is a closed subset in X whose lower box dimension is equal to r. The proved assertion makes it possible to strengthen the known results on intermediate values of the lower box dimension.