Subset sum into a consecutive range vs. standard subset sum
This answer assumes that a Yes instance of your problem with elements $x_0,\ldots,x_{n-1}$ and targets $[X,X+1]$ is one in which there is a subset summing to either $X$ or $X+1$.

Given an instance of your problem with elements $x_0,\ldots,x_{n-1}$ and targets $[X,X+1]$, construct an instance of subset sum with elements $x_0,\ldots,x_{n-1},1$ and target $X+1$. You take it from here.