This paper focuses on propagation phenomena in reaction-diffusion equations with a weakly monostable nonlinearity. The reaction term can be seen as an intermediate between the classical logistic one (or Fisher-KPP) and the standard weak Allee effect one. We investigate the effect of the decay rate of the initial data on the propagation rate. When the right tail of the initial data is sub-exponential, finite speed propagation and acceleration may happen and we derive the exact separation between the two situations. When the initial data is sub-exponentially unbounded, acceleration unconditionally occurs. Estimates for the locations of the level sets are expressed in terms of the decay of the initial data. In addition, sharp exponents of acceleration for initial data with sub-exponential and algebraic tails are given. Numerical simulations are presented to illustrate the above findings.