Dynamical mean-field theory is used to study the magnetic instabilities and phase diagram of the double-exchange (DE) model with Hund's coupling J_H > 0 in infinite dimensions. In addition to ferromagnetic (FM) and antiferromagnetic (AF) phases, the DE model also supports a broad class of short-range ordered (SRO) states with extensive entropy and short-range magnetic order. For any site on the Bethe lattice, the correlation parameter q of an SRO state is given by the average q = sin² (θ_i/2), where θ_i is the angle between any spin and its neighbours. Unlike the FM (q = 0) and AF (q = 1) transitions, the transition temperature of an SRO state with 0 < q < 1 cannot be obtained from the magnetic susceptibility. But a solution of the coupled Green's functions in the weak-coupling limit indicates that an SRO state always has a higher transition temperature than the AF for all fillings p below 1 and even has a higher transition temperature than the FM for 0.26 ≤ p ≤ 0.39. For 0.39 < p < 0.73, where both the FM and AF phases are unstable for small J_H, an SRO phase has a nonzero transition temperature except close to p = 0.5. As J_H increases, the SRO transition temperature eventually vanishes and the FM phase dominates the phase diagram. For small J_H, the T = 0 phase diagram of the DE model is greatly simplified by the presence of the SRO phase. An SRO phase is found to have lower energy than either the FM or AF phases for 0.26 ≤ p < 1. Phase separation (PS) disappears as J_H → 0 but appears for any nonzero coupling. For fillings near p = 1, PS occurs between an AF with p = 1 and either an SRO or a FM phase. The stability of an SRO state at T = 0 can be understood by examining the interacting density-of-states, which is gapped for any nonzero J_H in an AF but only when J_H exceeds a critical value in an SRO state.