When googling this problem myself, I found the highly-cited article
Terza, J.V., Basu, A. and Rathouz, P.J., 2008. Two-stage residual inclusion estimation: addressing endogeneity in health econometric modeling. Journal of health economics, 27(3), pp.531-543.
which proposes to use a method called 2-stage residual inclusion (2SRI) for the general linear model case. The method is very simple: Fit the first-stage model to get the residual and include both the residuals and the endogenous variable in the second-stage model.
Or more formally, let $𝑦_2$ be the endogenous variable, $𝑥_1$ till $𝑥_8$ the other exogenous control variables and $𝑖_1$ and $𝑖_2$ two instruments for $𝑦_2$. In the first stage, $𝑦_2$ is explained using linear regression
$𝑦_2=𝛼_0+𝛼_1 𝑥_1+𝛼_2 𝑥_2+…+𝛼_8 𝑥_8+𝛼_9 𝑖_1+𝛼_10 𝑖_2+𝜀_2$,
with $𝛼$ as coefficients and $𝜀_2$ as error term. The equation splits $𝑦_2$ in an exogenous component $𝛼_0+𝛼_1 𝑥_1+𝛼_2 𝑥_2+…+𝛼_8 𝑥_8+𝛼_9 𝑖_1+𝛼_10 𝑖_2$ and omitted-variable component $𝜀_2$. The 2SRI method includes both the endogenous variable $𝑦_2$ and the residual $𝜀_2$ as estimator for the omitted variable in the model; i.e. $𝑦_1=logit(𝛽_0+𝛽_1 𝑥_1+𝛽_2 𝑥_2+…+𝛽_8 𝑥_8+𝛽_9 𝑦_2+𝛽_10 𝜀_2 )+𝜀_1$
with $𝑦_1$ being the dichotomous variable. The implementation with a statics software is straight forward. (However, getting the standard errors for the estimators is not.)
It has been shown by
Burgess, S., & Thompson, S. G. (2012). Improving bias and coverage in instrumental variable analysis with weak instruments for continuous and binary outcomes. Statistics in medicine, 31(15), 1582-1600.
through a simulation that the 2SRI is better than 2SLS to provide another reference.