We investigate soliton self-compression and ultraviolet resonant dispersive wave emission in the higher-order modes of a gas-filled hollow capillary fibre (HCF). Our simple analytical scaling rules predict shorter required waveguides and different energy scales when moving from the fundamental to higher-order modes. Experimentally, we demonstrate soliton self-compression and ultraviolet dispersive wave emission in the double-lobe LP11 mode of an argon-filled HCF, which we excite by coupling into the fibre at oblique incidence. We observe the generation of ultraviolet dispersive waves which are frequency-shifted and more narrowband as compared to fundamental-mode generation due to the stronger modal dispersion, and a suppression of the supercontinuum between the dispersive wave and the pump pulse. With numerical simulations, we confirm the predictions of our scaling rules and find that the use of higher-order modes can suppress photoionisation and plasma effects even while allowing for much higher pulse energy to be used in the self-compression process. Our results add another degree of freedom for the design of hollow-waveguide systems to generate sub-cycle field transients and tuneable ultrashort laser pulses.