Core–non–penetrating molecular Rydberg states sample the electronic structure of the ion core in a uniquely simple way. However, radial overlap and orbital angular momentum propensity rules (Δℓ = ±1) can severely restrict spectroscopic access to ℓ >3 non–penetrating states, even via two–colour two–step excitation schemes. Owing to the profoundly different ℓ–uncoupling behaviour of moderate–n* (n*≈6–15) core–penetrating versus non–penetrating states, numerous and systematically repeated perturbations of the Hund's case (b) core–penetrating states accidentally located near integer n*) by the case (d) non–penetrating states always located near integer n*) can provide a valuable window into these information–rich, yet difficult to observe, core–non–penetrating states. We report analysis of core–penetrating (ℓ ≤ 3)–core–non–penetrating (ℓ > 3) perturbations in the fluorescence detected optical–optical double resonance spectrum of BaF. One of the perturbers of the 0.862Φ series is assigned as the lowest energy component, L = + 5, of an h complex (ℓ = 5). Other core–non–penetrating perturbers are assigned as components of g and h complexes. A critical aspect of the assignments of these quite weak and fragmentary interactions between one core–penetrating ‘bright’ state and many core–non–penetrating ‘dark’ states is the systematic (n* scaling of both matrix element and N values of the level crossing) repetition of each family of perturbations in each successive n* complex.