If I understand your question correctly the chain complex for $X[m]$ has i-chains the direct sum over partitions $P$ of (i-chains for $X$) tensor $U(i,m)(P)$ tensor $V(P)$ so that $C^i(X[m])=\oplus_P C^i(X)\otimes U^{i,m}_P\otimes V_P$ where $U(i,m)(P)$ has dimension $d(i,m)(P)$ which is the number of semistandard fillings of the shape $P$ with $0, 1, 2, \ldots, i, \ldots i$ and $V(P)$ is an irreducible representation of $S_m$.
For some representations this is easy to compute. For instance if $P(1)=m-d-1$ then $d(d-1,m)(P)$ is 0, but $d(d,m)(P)$ is nonzero so the $P$ representation in the top homology of $X[m]$ consists of (the d-chains of $X$) tensor $U(d,m)(P)$.