With a booming number of Type Ia supernovae (SNe Ia) discovered within a few days of their explosions, a fraction of SNe Ia that show luminosity excess in the early phase (early-excess SNe Ia) have been confirmed. In this article, we report early-phase observations of seven photometrically normal SNe Ia (six early detections and one deep non-detection limit) at the COSMOS field through a half-year transient survey as a part of the Hyper Suprime-Cam Subaru Strategic Program (HSC SSP). In particular, a blue light-curve excess was discovered for HSC17bmhk, a normal SN Ia with rise time longer than 18.8 days, during the first four days after the discovery. The blue early excess in optical wavelength can be explained not only by interactions with a non-degenerate companion or surrounding dense circumstellar matter but also radiation powered by radioactive decays of $^{56}$Ni at the surface of the SN ejecta. Given the growing evidence of the early-excess discoveries in normal SNe Ia that have longer rise times than the average and a similarity in the nature of the blue excess to a luminous SN Ia subclass, we infer that early excess discovered in HSC17bmhk and other normal SNe Ia are most likely attributed to radioactive $^{56}$Ni decay at the surface of the SN ejecta. In order to successfully identify normal SNe Ia with early excess similar to that of HSC17bmhk, early UV photometries or high-cadence blue-band surveys are necessary.