To change the phase in an FFT, first convert the complex output to r-θ format, modify the θ component, then change back to complex for the inverse FFT. The attached PNG should make this clearer. Note that the code in the PNG generates a time shift, so the phase linearly increases with frequency. As Mr. Pursley made clear, you should keep time shifts to a minimum to avoid too much ringing at sharp edges. I use this to tweak a data set a small increment (less than a sample width). For larger increments, I simply take a subset of the data, then use this technique to move the last small bit.
