  We study Gabor frames with Hermite window functions. Gr\"ochenig and
Lyubarskii provided a sufficient density condition for their frame sets, which
leads to what we call the ``safety region". For rectangular lattices and
Hermite windows of order 4 and higher, we enlarge this safety region by
providing new points on the boundary of this region. For this purpose, we
employ the Janssen representation of the frame operator to compare its distance
to the identity in the operator norm. The calculations lead to estimates on
series involving Laguerre polynomials with Gaussian weight functions.
