We consider the N-dimensional Cauchy problem in R N for a semilinear damped wave equation with a power-type nonlinearity | u | p . For a noncompactly supported initial data, which has a small energy, we shall derive a global in time existence result in the case when the power of the nonlinear term satisfies 1 + 2 / N < p ⩽ N / [ N - 2 ] + . This generalizes a previous result due to Todorova–Yordanov (J. Differential Equations 174 (2001) 464–489), which dealt with a solution in the framework of compactly supported initial data.