  We set up a new, flexible approach for the tensor reduction of one-loop
Feynman integrals. The 5-point tensor integrals up to rank R=5 are expressed by
4-point tensor integrals of rank R-1, such that the appearance of the inverse
5-point Gram determinant is avoided. The 4-point tensor coefficients are
represented in terms of 4-point integrals, defined in $d$ dimensions,
$4-2\epsilon \le d \le 4-2\epsilon+2(R-1)$, with higher powers of the
propagators. They can be further reduced to expressions which stay free of the
inverse 4-point Gram determinants but contain higher-dimensional 4-point
integrals with only the first power of scalar propagators, plus 3-point tensor
coefficients. A direct evaluation of the higher dimensional 4-point functions
would avoid the appearance of inverse powers of the Gram determinants
completely. The simplest approach, however, is to apply here dimensional
recurrence relations in order to reduce them to the familiar 2- to 4-point
functions in generic dimension $d = 4-2\eps$, introducing thereby coefficients
with inverse 4-point Gram determinants up to power $R$ for tensors of rank $R$.
For small or vanishing Gram determinants - where this reduction is not
applicable - we use analytic expansions in positive powers of the Gram
determinants. Improving the convergence of the expansions substantially with
Pad\'e approximants we close up to the evaluation of the 4-point tensor
coefficients for larger Gram determinants. Finally, some relations are
discussed which may be useful for analytic simplifications of Feynman diagrams.
