We combine multiple semicoarsened multigrid (MSG) with unbiased multi-index Monte Carlo (MIMC) and show how coarse solutions from the MSG solver can be reused in the estimator. We analyze the cost of this new estimator theoretically and present numerical experiments with various anisotropic random fields, where the unknown coefficients in the covariance model are considered as hyperparameters. We illustrate its robustness and superiority over unbiased MIMC without sample reuse.