  Let $X$ be a smooth complex projective curve, and let $E$ be a vector bundle
on $X$ which is not semistable. For a suitably chosen integer $r$, let
$\text{Gr}(E)$ be the Grassmann bundle over $X$ that parametrizes the quotients
of the fibers of $E$ of dimension $r$. Assuming some numerical conditions on
the Harder-Narasimhan filtration of $E$, we study Seshadri constants of ample
line bundles on $\text{Gr}(E)$. In many cases, we give the precise value of
Seshadri constant. Our results generalize various known results for ${\rm
rank}(E)=2$.
