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We generalize the example of linear set presented by the last two authors in"Vertex properties of maximum scattered linear sets of $\mathrm{PG}(1,q^n)$"(2019) to a more general family, proving that such linear sets are maximumscattered when $q$ is odd and, apart from a special case, they are are new.This solves an open problem posed in "Vertex properties of maximum scatteredlinear sets of $\mathrm{PG}(1,q^n)$" (2019). As a consequence of Sheekey'sresults in "A new family of linear maximum rank distance codes" (2016), thisfamily yields to new MRD-codes with parameters $(6,6,q;5)$.

A new family of maximum scattered linear sets in $\mathrm{PG}(1,q^6)$