  In this paper, let $\mathcal{S}$ denote the possible interior singular set of
suitable weak solutions of the 3D Navier-Stokes equations. We improve the known
upper box-counting dimension of this set from $360/277(\approx1.300)$ in [24]
to $975/758(\approx1.286)$. It is also shown that
$\Lambda(\mathcal{S},r(\log(e/r))^{\sigma})=0(0\leq\sigma<27/113)$, which
extends the previous corresponding results concerning the improvement of the
classical Caffarelli-Kohn-Nirenberg theorem by a logarithmic factor in Choe and
Lewis [3, J. Funct. Anal., 175: 348-369, 2000] and in Choe and Yang et al. [4,
Comm. Math. Phys, 336: 171-198, 2015]. The proof is inspired by a new
$\varepsilon$-regularity criterion proved by Guevara and Phuc in [7, Calc. Var.
56:68, 2017].
