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We demonstrate low-threshold single-mode supercontinuum (SC) generation by using a tellurite nanowire with long suspended struts under the pump of a picosecond fiber laser. The nanowire was fabricated by pumping an inflation pressure of nitrogen gas into the holes in the processes of preparing cane and drawing nanowire. At the pump wavelength of 1064 nm the nanowire has an optimized nonlinear coefficient. Due to the long radial struts the nanowire can propagate light up to the wavelength as long as 1800 nm with an insignificant confinement loss. Additionally, the tube-shaped glass cladding of the suspended nanowire shifts the single-mode cutoff wavelength to 810 nm. It is shorter than the cutoff wavelength, 1070 nm, of a naked nanowire with the same diameter. This makes it possible to obtain an SC spectrum, which is single mode for all wavelengths of the spectrum. Because of the high nonlinearity, a single-mode SC covering a wavelength range of 900-1600 nm was obtained under the pump pulse with the peak power as low as 24 W. It is shown that the suspended core nanowire with long struts can provide an optimized nonlinearity together with single-mode propagation, which leads to interesting applications in compact nonlinear devices.