  In the rank modulation scheme, Gray codes are very useful in the realization
of flash memories. For a Gray code in this scheme, two adjacent codewords are
obtained by using one "push-to-the-top" operation. Moreover, snake-in-the-box
codes under the $\ell_{\infty}$-metric are Gray codes, which can be capable of
detecting one $\ell_{\infty}$-error. In this paper, we give two constructions
of $\ell_{\infty}$-snakes. On the one hand, inspired by Yehezkeally and
Schwartz's construction, we present a new construction of the
$\ell_{\infty}$-snake. The length of this $\ell_{\infty}$-snake is longer than
the length of the $\ell_{\infty}$-snake constructed by Yehezkeally and
Schwartz. On the other hand, we also give another construction of
$\ell_{\infty}$-snakes by using $\mathcal{K}$-snakes and obtain the longer
$\ell_{\infty}$-snakes than the previously known ones.
