  For a non-decreasing sequence $S=(s_1,s_2,\ldots)$ of positive integers, a
partition of the vertex set of a graph $G$ into subsets $X_1,\ldots, X_\ell$,
such that vertices in $X_i$ are pairwise at distance greater than $s_i$ for
every $i\in\{1,\ldots,\ell\}$, is called an $S$-packing $\ell$-coloring of $G$.
The minimum $\ell$ for which $G$ admits an $S$-packing $\ell$-coloring is
called the $S$-packing chromatic number of $G$, denoted by $\chi_S(G)$. In this
paper, we consider $S$-packing colorings of distance graphs
$G(\mathbb{Z},\{k,t\})$, where $k$ and $t$ are positive integers, which are the
graphs whose vertex set is $\mathbb{Z}$, and two vertices $x,y\in \mathbb{Z}$
are adjacent whenever $|x-y|\in\{k,t\}$. We complement partial results from two
earlier papers, thus determining all values of $\chi_S(G(\mathbb{Z},\{k,t\}))$
when $S$ is any sequence with $s_i\le 2$ for all $i$. In particular, if
$S=(1,1,2,2,\ldots)$, then the $S$-packing chromatic number is $2$ if $k+t$ is
even, and $4$ otherwise, while if $S=(1,2,2,\ldots)$, then the $S$-packing
chromatic number is $5$, unless $\{k,t\}=\{2,3\}$ when it is $6$; when
$S=(2,2,2,\ldots)$, the corresponding formula is more complex.
