Low-frequency combustion instabilities arise in an aero-engine gas turbine combustor under idle conditions, particularly when pilot-mode non-premixed combustion is sustained. Despite the fundamental importance of such instabilities, little is known about how they are initiated and grow in the system. Our results demonstrate that a non-premixed Jet A-1 spray flame yields an intermediate-amplitude, quasi-periodic L1 mode oscillation for a low pilot equivalence ratio, and the frequency gradually increases with increasing fuel flowrates. Subsequent to a critical point, the system undergoes a discontinuous mode transition, giving rise to the formation of an extremely large pressure oscillation with an L2 mode structure. Several key triggers were found to induce the mode shift: (i) generation of a high intensity pulse in the flame's heat release rate due to the spontaneous ignition of unburned reactant mixtures, (ii) emergence of a large-scale vortical structure and its interaction with a partially premixed flame front, and (iii) development of self-sustained limit cycle oscillations driven by periodic convection of a hot spot. Our study identifies the occurrence of a large-amplitude peak followed by a local minimum intensity, analogous to the activation energy concept, as an essential step for entry into a new state with large-amplitude limit cycle oscillations.