Unamplified subnanojoule 30-fs pulses of Cr:forsterite laser radiation are shown to generate the third-harmonic signal in a tapered fiber. Analysis of the dispersion properties of a tapered fiber indicates the possibility of phase matching third-harmonic generation in higher-order waveguide modes. When the pump and third-harmonic pulses are group-velocity-mismatched, optimal phase matching is achieved for frequency components spectrally offset from the central frequency of the pump pulse, which gives rise to an asymmetry in the spectrum of the third harmonic.