Voice anonymisation aims to conceal speaker identity while preserving linguistic content. Most approaches target predominantly timbral cues and often overlook non-timbral cues such as prosody, rhythm, speaking style and accent, which may still leak speaker-specific information related to voice identity after anonymisation. With this paper, we propose a speaker anonymisation system that explicitly obfuscates both timbral and non-timbral cues. Extensive experiments conducted within the VoicePrivacy Challenge framework show improved protection against attacks exploiting non-timbral information compared to state-of-the-art systems. For evaluation, we use a pair of complementary automatic speaker verification models to demonstrate improved anonymisation robustness by 32% relative to attacks which target either timbral and non-timbral cues. Results also show stronger anonymisation comes at the cost of only moderate degradation to intelligibility and naturalness.