This paper investigates the dynamical connection function $\gamma(t)$ in $f(Q)$ cosmology, using Hubble data and Gaussian Processes to reconstruct it beyond the coincident gauge. The authors explore three branches of connections that satisfy the torsionless and curvatureless conditions, parameterized by $\gamma$. They express the redshift dependence of $\gamma$ in terms of the $H(z)$ function and the $f(Q)$ form and parameters, and then reconstruct it using 55 $H(z)$ observation data. The study is divided into two main parts: 1. **Reconstruction for the Matter Conservation Case**: In this part, the authors assume the ordinary conservation law for the matter sector. They derive a general solution for the derivative of $f(Q)$ and reconstruct the redshift dependence of $\gamma(z)$. The reconstructed $f(Q)$ function is well described by a quadratic correction on top of Symmetric Teleparallel Equivalent of General Relativity (STEGR). 2. **Reconstruction in the General Case**: In this part, the authors consider two well-studied $f(Q)$ models: the square-root (Sqrt-$f(Q)$) and exponential (Exp-$f(Q)$) models. They reconstruct $\gamma(z)$ for both models and show that the data favor a non-zero $\gamma(z)$. The inclusion of $\gamma$ improves the quality of the fittings compared to the $\Lambda$CDM paradigm and the same $f(Q)$ models under the coincident gauge. The results indicate that $f(Q)$ cosmology should be studied beyond the coincident gauge, as the inclusion of $\gamma$ enhances the fit to observational data. The paper concludes with a discussion on the implications of these findings and suggests future directions for further research, including the use of different observational datasets and the examination of perturbation evolution.This paper investigates the dynamical connection function $\gamma(t)$ in $f(Q)$ cosmology, using Hubble data and Gaussian Processes to reconstruct it beyond the coincident gauge. The authors explore three branches of connections that satisfy the torsionless and curvatureless conditions, parameterized by $\gamma$. They express the redshift dependence of $\gamma$ in terms of the $H(z)$ function and the $f(Q)$ form and parameters, and then reconstruct it using 55 $H(z)$ observation data. The study is divided into two main parts: 1. **Reconstruction for the Matter Conservation Case**: In this part, the authors assume the ordinary conservation law for the matter sector. They derive a general solution for the derivative of $f(Q)$ and reconstruct the redshift dependence of $\gamma(z)$. The reconstructed $f(Q)$ function is well described by a quadratic correction on top of Symmetric Teleparallel Equivalent of General Relativity (STEGR). 2. **Reconstruction in the General Case**: In this part, the authors consider two well-studied $f(Q)$ models: the square-root (Sqrt-$f(Q)$) and exponential (Exp-$f(Q)$) models. They reconstruct $\gamma(z)$ for both models and show that the data favor a non-zero $\gamma(z)$. The inclusion of $\gamma$ improves the quality of the fittings compared to the $\Lambda$CDM paradigm and the same $f(Q)$ models under the coincident gauge. The results indicate that $f(Q)$ cosmology should be studied beyond the coincident gauge, as the inclusion of $\gamma$ enhances the fit to observational data. The paper concludes with a discussion on the implications of these findings and suggests future directions for further research, including the use of different observational datasets and the examination of perturbation evolution.