  P.B. Chakraborty {\it et al.}, Phys. Rev. B {\bf 70}, 144411 (2004)) study of
the LiHoF$_4$ Ising magnetic material in an external transverse magnetic field
$B_x$ show a discrepancy with the experimental results, even for small $B_x$
where quantum fluctuations are small. This discrepancy persists asymptotically
close to the classical ferromagnet to paramagnet phase transition. In this
paper, we numerically reinvestigate the temperature $T$, versus transverse
field phase diagram of LiHoF$_4$ in the regime of weak $B_x$. In this regime,
starting from an effective low-energy spin-1/2 description of LiHoF$_4$, we
apply a cumulant expansion to derive an effective temperature-dependent
classical Hamiltonian that incorporates perturbatively the small quantum
fluctuations in the vicinity of the classical phase transition at $B_x=0$. Via
this effective classical Hamiltonian, we study the $B_x-T$ phase diagram via
classical Monte Carlo simulations. In particular, we investigate the influence
on the phase diagram of various effects that may be at the source of the
discrepancy between the previous QMC results and the experimental ones. For
example, we consider two different ways of handling the long-range
dipole-dipole interactions and explore how the $B_x-T$ phase diagram is
modified when using different microscopic crystal field Hamiltonians. The main
conclusion of our work is that we fully reproduce the previous QMC results at
small $B_x$. Unfortunately, none of the modifications to the microscopic
Hamiltonian that we explore are able to provide a $B_x-T$ phase diagram
compatible with the experiments in the small semi-classical $B_x$ regime.
