  We theoretically clarify the functional form to be used in $t \to 0$
extrapolation in the small flow time expansion (SF$t$X) method for the
energy-momentum tensor (EMT), which facilitates lattice simulation of the EMT
based on the gradient flow. We argue that in the $t \to 0$ extrapolation
analysis, lattice data should be fitted by a power function in $g(\mu(t))$, the
flow time dependent running coupling, where the power is determined by the
perturbation order we consider. From actual lattice data, we confirm the
validity of the extrapolation function. Using the new extrapolation function,
we present updated lattice results for thermodynamics quantities in quenched
QCD; our results are consistent with the previous study [arXiv:1812.06444] but
we obtain smaller errors due to reduction of systematic errors.
