When a cylinder is mounted on an elastic support within a current, vortex-induced vibrations (VIV) may occur down to a Reynolds number (Re) close to 20, based on the body diameter (D) and inflow velocity (U), i.e. below the critical value of 47 reported for the onset of flow unsteadiness when the body is fixed. The impact of a forced rotation of the elastically mounted cylinder on the system behaviour is explored numerically for Re <= 30, over wide ranges of values of the rotation rate (ratio between body surface velocity and U, alpha is an element of [0, 5]) and reduced velocity (inverse of the oscillator natural frequency non-dimensionalized by D and U, U* is an element of [2, 30]). The influence of the rotation is not monotonic, but the most prominent effect uncovered in this work is a substantial enhancement of the subcritical-Re, flow-induced vibrations beyond alpha = 2. This enhancement is twofold. First, the rotation results in a considerable expansion of the vibration/flow unsteadiness region in the (Re, U*) domain, down to Re = 4. Second, the elliptical orbits described by the rotating body are subjected to a major amplification, with a transition from VIV to responses whose magnitude tends to increase unboundedly with U*, even though still synchronized with flow unsteadiness. The emergence of such galloping-like oscillations close to the onset of vibrations disrupts the scenario of gradual vibration growth with Re, as amplitudes larger than 10 body diameters may be observed at Re = 10.