Broadband supercontinuum (SC) light sources generated through nonlinear effects in solid-core photonic crystal fibers (PCFs) have been widely used in spectroscopy, metrology, and microscopy, leading to great application successes. The short-wavelength extension of such SC sources, a longstanding challenge, has been the subject of intensive study over the past two decades. However, the exact mechanism of blue and ultraviolet light generation, especially for some resonance spectral peaks in the short-wavelength regime, is not yet fully understood. Here, we demonstrate that the effect of inter-modal dispersive-wave radiation, which results from phase matching between pump pulses at the fundamental optical mode and packets of linear waves at some higher-order modes (HOMs) propagating in the PCF core, might be one of the critical mechanisms that can result in some resonance spectral components with wavelengths much shorter than that of the pump light. We observed in an experiment that several spectral peaks resided in the blue and ultraviolet regimes of the SC spectrum, whose central wavelengths can be tuned by varying the PCF-core diameter. These experimental results can be interpreted well using the inter-modal phase-matching theory, providing some useful insights into the SC generation process.