Solving fluid-structure interaction (FSI) problems when the fluid and structure densities are similar (large added mass), such as in hemodynamics, is challenging since the stability and convergence properties of the adopted numerical scheme could be compromised (Causin et al., CMAME, 2005). This is especially true for partitioned schemes (Foster et al., CMAME, 2007; van Brummelen et al., J. Appl. Math., 2009), whose modularity is otherwise attractive, allowing for better-conditioned subproblems and the exploitation of existing standalone fluid and structural codes. Building on the interpretation of the standard Dirichlet-Neumann (DN) algorithm as a Richardson method with a block Gauss-Seidel preconditioner and acceleration parameter α = 1, we develop new coupling strategies suited for large added-mass regimes. Specifically, we consider the strongly-coupled (SC) method associated with Richardson, using optimal values of α (SC-DN-α algorithm). The resulting scheme introduces correction terms that improve the convergence of the standard DN method. We analytically prove that, for the linear model problem proposed by (Causin et al., CMAME, 2005), the SC-DN-α method is convergent over a specific range of α values without requiring relaxation, even in the presence of large added mass. Starting from this formulation, we derive a new loosely-coupled (LC) scheme (LC-DN-α) obtained by performing only one preconditioned Richardson iteration per time step. Stability analysis on a benchmark problem proves that the proposed LC scheme is conditionally stable in large added mass regimes, provided a constraint on α is met. We then study how the stability region depends on the spatial and temporal discretization parameters and introduce a practical strategy for selecting stable values of α. 2D numerical experiments in hemodynamic settings confirm the theoretical results, demonstrating the effectiveness of the proposed scheme and highlighting its potential as a prototype for robust preconditioning strategies with respect to physical parameters.