  In [10], the existence of $\mathbb{F}_q$-linear MRD-codes of
$\mathbb{F}_q^{6\times 6}$, with dimension $12$, minimum distance $5$ and left
idealiser isomorphic to $\mathbb{F}_{q^6}$, defined by a trinomial of
$\mathbb{F}_{q^6}[x]$, when $q$ is odd and $q\equiv 0,\pm 1\pmod 5$, has been
proved. In this paper we show that this family produces $\mathbb{F}_q$-linear
MRD-codes of $\mathbb{F}_q^{6\times 6}$, with the same properties, also in the
remaining $q$ odd cases, but not in the $q$ even case. These MRD-codes are not
equivalent to the previously known MRD-codes. We also prove that the
corresponding maximum scattered $\mathbb{F}_q$-linear sets of
$\mathrm{PG}(1,q^6)$ are not $\mathrm{P}\Gamma\mathrm{L}(2,q^6)$-equivalent to
any previously known scattered linear set.
