The interaction between the magnetic moment, due to the spin of the nucleus, and the larger magnetic moment, due to the electron's spin, results in energy shifts which are much smaller than those of the fine structure. They are of order and are hence called hyperfine.
The hyperfine corrections may be difficult to measure in transitions between states of different , however, they are quite measurable and important because they split the ground state. The different hyperfine levels of the ground state are populated thermally. Hyperfine transitions, which emit radio frequency waves, can be used to detect interstellar gas.
This material is covered in Gasiorowicz Chapter 17, in Cohen-Tannoudji et al. Chapter XII,and briefly in Griffiths 6.5.