An $L(2,1)$-labelling of a graph $G$ is an assignment of non-negative integers to the vertices of $G$ such that vertices at distance at most two get different numbers and adjacent vertices get numbers which are at least two apart. The $L(2,1)$-labelling number of $G$, denoted by $\lambda(G)$, is the minimum range of labels over all such labellings. In this paper, we first discuss some necessary and sufficient conditions for unit interval graph $G$ to have $\lambda(G)=2\chi(G)-2$ and then characterize all unit interval graphs $G$ of order no more than $3\chi(G)-1$, where $\chi(G)$ is the chromatic number of $G$. Finally, we discuss some subgraphs of unit interval graphs $G$ on more than $2\chi(G)+1$ vertices with $\lambda(G) = 2\chi(G)$.