We complete the mathematical analysis of the fine structure of harmonic measure on SLE curves that was initiated in Ref. [2], as described by the averaged integral means spectrum. For the unbounded version of whole-plane SLE as studied in Refs. [7, 16], a phase transition has been shown to occur for high enough moments from the bulk spectrum towards a novel spectrum related to the point at infinity. For the bounded version of whole-plane SLE of Ref. [2], a similar transition phenomenon, now associated with the SLE origin, is proved to exist for low enough moments, but we show that it is superseded by the earlier occurrence of the transition to the SLE tip spectrum.