Moduli fields with Planck-suppressed couplings to light species are common in string compactifications. Decays of these moduli can reheat the universe at a late time and produce dark matter non-thermally. For generic moduli fields motivated by string theory with masses similar to that of the gravitino and TeV-scale superpartners in the minimal supersymmetric Standard Model (MSSM), the non-thermal production of the lightest superpartner (LSP) tends to create an unacceptably large relic density or too strong of an indirect detection signal. We call this the moduli-induced LSP problem of the MSSM. In this paper we investigate extensions of the MSSM containing new LSP candidates that can alleviate this tension. We examine the viability of this scenario in models with light Abelian and non-Abelian hidden sectors, and symmetric or asymmetric dark matter. In these extensions it is possible, though somewhat challenging, to avoid a moduli-induced LSP problem. In all but the asymmetric scenario, the LSP can account for only a small fraction of the observed dark matter density.