The Landau-Wilson field theory with O(n)×O(m) symmetry which describes the critical thermodynamics of frustrated spin systems with noncollinear and noncoplanar ordering is analyzed in 4−ε dimensions within the minimal subtraction scheme in the six-loop approximation. The ε expansions for marginal dimensionalities of the order parameter n^H(m,4−ε), n⁻(m,4−ε), n⁺(m,4−ε) separating different regimes of critical behavior are extended up to ε⁵ terms. Concrete series with coefficients in decimals are presented for m={2,…,6}. The diagram of stability of nontrivial fixed points, including the chiral one, in (m,n) plane is constructed by means of summing up of corresponding ε expansions using various resummation techniques. Numerical estimates of the chiral critical exponents for several couples {m,n} are also found. Comparative analysis of our results with their counterparts obtained earlier within the lower-order approximations and by means of alternative approaches is performed. It is confirmed, in particular, that in physically interesting cases n=2,m=2 and n=2,m=3 phase transitions into chiral phases should be first-order.