  Critical exponent $\eta$ for three-dimensional systems with $n$-vector order
parameter is evaluated in the frame of pseudo-$\epsilon$ expansion approach.
Pseudo-$\epsilon$ expansion ($\tau$-series) for $\eta$ found up to $\tau^7$
term for $n$ = 0, 1, 2, 3 and within $\tau^6$ order for general $n$ is shown to
have a structure rather favorable for getting numerical estimates. Use of
Pad\'e approximants and direct summation of $\tau$-series result in iteration
procedures rapidly converging to the asymptotic values that are very close to
most reliable numerical estimates of $\eta$ known today. The origin of this
fortune is discussed and shown to lie in general properties of the
pseudo-$\epsilon$ expansion machinery interfering with some peculiarities of
the renormalization group expansion of $\eta$.
