  We apply the $k_T$-factorization approach to the production of $\phi$ meson
in deep-inelastic scattering. The helicity-conserving $\gamma^*(T,L) \to \phi$
impact factor is calculated for longitudinal and transverse photon polarization
using $\phi$ meson distribution amplitudes. Different unintegrated gluon
distributions are used in the calculations. The formalism for massless
quarks/antiquarks gives too large transverse and longitudinal cross sections
for photon virtualities below $Q^2\sim 8 \, \rm{GeV}^2$. We suggest how to
improve the description of the HERA data by introducing effective strange quark
masses into the formalism. We derive the corresponding massive impact factor by
comparing to the light-cone wave function representation used in previous
$k_T$-factorization calculations and the color-dipole approaches. As a
byproduct we present expressions for higher twist-amplitudes as weighted
integrals over the light-cone wave function. The quark mass $m_q \approx 0.5$
GeV allows to improve the description of both longitudinal and transverse cross
section down to $Q^2 \sim$ 4 GeV$^2$. We present also the polarized cross
section ratio $\sigma_L/\sigma_T$ and the behavior of the total cross section
$\sigma_{tot} = \sigma_{L} + \sigma_{T}$ as a function of photon virtuality.
