Kagom{e}-staircase compound Ni$_3$V$_2$O$_8$ is an attractive multiferroic material exhibiting rich phase diagrams. However, the magnetic properties and magnetic transitions have been studied only above 1.3 K. In this work, we study the thermal conductivity ($kappa$) of Ni$_3$V$_2$O$_8$ single crystals at low temperatures down to 0.3 K and in magnetic fields up to 14 T. In zero field, the magnetic transitions from the low-temperature incommensurate (LTI) phase to the commensurate phase (C) and then to a second commensurate phase (C) yield anomalies in $kappa(T)$ curves at $Trm_{LC}$ = 3.7 K and $Trm_{CC}$ = 2.0 K, respectively, which indicates a significant phonon scattering by the critical spin fluctuations. When the field is applied along the $a$ axis, the field dependence of $kappa$ displays four anomalies associated with different magnetic transitions and reveals an undetected magnetic state at subKelvin temperatures. In addition, the $kappa(B)$ curves are found to depend not only on the history but also on the magnitude of applying field. When the field is applied along the $b$ axis, a high-field phase locating above the LTI and high-temperature incommensurate (HTI) phases is revealed.
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