An $[n,k,d]$ linear code is said to be maximum distance separable (MDS) or almost maximum distance separable (AMDS) if $d=n-k+1$ or $d=n-k$, respectively. If a code and its dual code are both AMDS, then the code is said to be near maximum distance separable (NMDS). For $k=3$ and $k=4$, there are many constructions of NMDS codes by adding some suitable projective points to arcs in $\mathrm{PG}(k-1,q)$. In this paper, we consider the monomial equivalence problem for some NMDS codes with the same weight distributions and present new constructions of NMDS codes.