This study experimentally investigates the effects of nose-tip bluntness on boundary-layer transition over a \(7^\circ\) half-angle cone at Mach 6.76, using the Seoul National University Hypersonic Shock Tunnel. Transition characteristics were examined through high-speed schlieren visualization, surface heat flux measurements, and high-frequency surface pressure measurements for varying nose-tip radii (0.1 mm, 1 mm, and 2 mm) and unit Reynolds numbers. Increasing nose-tip bluntness effectively delayed transition onset, as indicated by turbulent intermittency and heat flux distributions. Spectral proper orthogonal decomposition and pressure spectral analyses revealed distinct second-mode instabilities with frequency shifts to lower values as bluntness increased. Additionally, a low-frequency instability around 200 kHz was identified in the configuration with a 2 mm nose-tip radius, suggesting the presence of multiple instability modes. These observations highlight the influence of nose-tip bluntness on hypersonic boundary-layer stability and emphasize the necessity for comprehensive consideration of multiple instability modes in hypersonic vehicle design.