The effect of fluctuating (nonpreaveraged) hydrodynamic interaction (HI) on the hydrodynamic-radius expansion factor alpha(H) for the Gaussian chain with excluded volume is theoretically examined in the uniform-expansion approximation within the framework of the theory previously developed for the unperturbed Gaussian chain with fluctuating HI. It is shown that alpha(H) may be given by the product of the Zimm hydrodynamic-radius expansion factor alpha(H)((Z)) and a newly introduced factor, which represents the present effect of fluctuating HI on alpha(H) and is generally smaller than unity. Thus it is concluded that the fluctuating HI decreases alpha(H) from alpha(H)((Z)). An approximate interpolation formula for alpha(H) is also constructed as a function of the conventional excluded-volume parameter z and is completed by replacing z by the scaled excluded-volume parameter ($) over bar z in the quasi-two-parameter scheme.