We provide quantitative inner and outer bounds for the symmetric quasiconvex hull \(Q^e({\mathcal {U}})\) on linear strains generated by three-well sets \({\mathcal {U}}\) in \({\mathbb {R}}^{2\times 2}_{sym}\). In our study, we consider all possible compatible configurations for three wells and prove that if there exist two matrices in \({\mathcal {U}}\) that are rank-one compatible then \(Q^e({\mathcal {U}})\) coincides with its symmetric lamination convex hull \(L^e({\mathcal {U}})\). We complete this result by providing an explicit characterization of \(L^e({\mathcal {U}})\) in terms of the wells in \({\mathcal {U}}\). Finally, we discuss the optimality of our outer bound and its relationship with quadratic polyconvex functions.