  We propose a new approximation scheme to solve the Non Perturbative
Renormalization Group equations and obtain the full momentum dependence of
$n$-point functions. This scheme involves an iteration procedure built on an
extension of the Local Potential Approximation commonly used within the Non
Perturbative Renormalization Group. Perturbative and scaling regimes are
accurately reproduced. The method is applied to the calculation of the shift
$\Delta T_c$ in the transition temperature of the weakly repulsive Bose gas, a
quantity which is very sensitive to all momenta intermediate between these two
regions. The leading order result is in agreement with lattice calculations,
albeit with a theoretical uncertainty of about 25%. The next-to-leading order
differs by about 10% from the best accepted result.
