Goerzel algorithm - increasing accuracy
There are two things you can do:


Use a different algorithm, like autocorrelation.
Use the phase information! This technique is discussed in chapter 9 of DAFX. You can find matlab source here.


DAFX discussed the phase-based technique using the FFT, but it should apply to Goerzel's algorithm as well, assuming you have real data at that frequency. The key concept is that you must note how much the phase changes over time to determine the exact frequency.

You may want to experiment with an FFT of smaller size instead of Goerzel's algorithm to validate the assumptions.