We calculate the PCAC mass for $(2+1)$ flavor full QCD with Wilson-type quarks. We adopt the \textbf{S}mall \textbf{F}low-\textbf{\textit{t}}ime e\textbf{X}pansion (\textbf{SF\textit{t}X}) method based on the gradient flow which provides us a general way to compute correctly renormalized observables even if the relevant symmetries for the observable are broken explicitly due to the lattice regularization, such as the Poinc\'{a}re and chiral symmetries. Our calculation is performed on heavy $u, d$ quarks mass ($m_{\pi}/m_{\rho}\simeq0.63$) and approximately physical $s$ quark mass with fine lattice $a \simeq 0.07$~fm. The results are compared with those computed with the Schr\"odinger functional method.