Trying to decompose a signal into sine waves whose wavelengths aren't restricted to 1/N times sample range?
There is absolutely no need for explicit interpolation. The DFT can do that for you. Take you signal frame and apply a windowing function to minimise edge artefacts. Afterwards pad the frame with zeros to determine the number to output bins of the DFT. Then apply your FFT and you have an interpolated spectrum with the frequency step size you desire.