The Nosé-Hoover (NH) equation of motion is widely used in molecular dynamics simulations. It enables us to set a constant temperature and produce the canonical distribution for a target physical system. For the purpose of investigating the physical system under fluctuating temperature, we have introduced a coupled Nosé-Hoover equation in our previous work [J. Phys. A 48 455001 (2015)]. The coupled NH equation implements a fluctuating heat-bath temperature in the NH equation of the physical system, and also keeps a statistically complete description via an invariant measure of the total system composed of the physical system and a "temperature system". However, a difficulty lies in that the time development of the physical system may not correspond to the realistic physical process, because of the need of a scaled time average to compute thermodynamical quantities. The current work gives a solution by presenting a new scheme, which is free from the scaled time but retains the statistical description. By use of simple model systems, we validate the current scheme and compare with the original scheme. The sampling property of the current scheme is also clarified to investigate the effect of function setting used for the distribution of the total system.