  We construct torsion-free hyperbolic groups without unique product whose
subgroups up to some given finite index are themselves non-unique product
groups. This is achieved by generalising a construction of Comerford to
graphical small cancellation presentations, showing that for every subgroup $H$
of a graphical small cancellation group there exists a free group $F$ such that
$H*F$ admits a graphical small cancellation presentation.
