  Studying the upper critical field ($\mu_0$$H$$_{\rm{c2}}$) and its anisotropy
of superconductors is of great importance because it can provide an unusual
insight into the pair-breaking mechanism. Since Fe$_{1+y}$Te$_{1-x}$Se$_x$
exhibits the high $\mu_0$$H$$_{\rm{c2}}$ and small anisotropic
superconductivity, it has attracted considerable attention. However, some
issues related to $\mu_0$$H$$_{\rm{c2}}$ are still unknown, including the
effect of excess Fe content on $\mu_0$$H$$_{\rm{c2}}$ behavior and the origin
of the crossover of the $\mu_0H_{\rm{c2}}^c $ -- $ T$ and
$\mu_0H_{\rm{c2}}^{ab}$ -- $T$ curves. In this work, the value of
$\mu_0$$H$$_{\rm{c2}}$ of Fe$_{1+y}$Te$_{0.6}$Se$_{0.4}$ single crystals with
controlled amounts of excess Fe was obtained by resistivity measurements over a
wide range of temperatures down to $\sim$ 1.5 K, and magnetic fields up to
$\sim$ 60 T. The crossover of the $\mu_0H_{\rm{c2}}^c $ -- $ T$ and
$\mu_0H_{\rm{c2}}^{ab}$ -- $T$ curves was found to be independent of the excess
Fe content. The angle dependence of $\mu_0H_{\rm{c2}}$ was also checked. The
$\mu_0H_{\rm{c2}}(\theta)$ symmetry at higher temperature near $T_c$ could be
fitted by anisotropic G-L model, and novel fourfold symmetry of
$\mu_0H_{\rm{c2}}$ at lower temperature was found. Based on our spin-locking
pairing model, the crossover behavior originates from the anisotropic
spin-paramagnetic effect, and the novel fourfold symmetry of $\mu_0H_{\rm{c2}}$
could be understood by our extended anisotropic G-L model.
