If $x_i=i/m$ for $i=1,\dots,m-1$, this is possible to do for $m=2,3,4$ but apparently not for $m=5$.
Indeed, it appears that for $m=5$ and for all natural $n$ $$d_{m,n}:=d(J_m^{(n)}, \{i/m\}_{i=1}^{m-1}) =\frac2{mn}. \tag{1}\label{1}$$
To illstrate this, below is the (connected) graph $\{(n,m\, n\, d_{m,n})\colon n=1,\dots,100\}$ for $m=5$:
It should not be hard to prove \eqref{1} for $m=5$.