We prove new results on single point Seshadri constants for ample line bundles on hyperelliptic surfaces, motivated by the results of Farnik [Arch. Math. 107 (2016), pp. 227–237]. Given a hyperelliptic surface
X
X
and an ample line bundle
L
L
on
X
X
, we show that the least Seshadri constant
ε
(
L
)
\varepsilon (L)
of
L
L
is a rational number when
X
X
is not of type 6. We also prove new lower bounds for the Seshadri constant
ε
(
L
,
1
)
\varepsilon (L,1)
of
L
L
at a very general point.