In life histories with generation overlap, selection that acts differently on different life-stages can produce reservoirs of genetic variation, for example in long-lived dormant propagules. Such reservoirs provide "migration from the past" to the current population, and they have the potential either to slow adaptive evolution or accelerate it by re-introducing genotypes not affected by recent selection (e.g., through storage effect in a fluctuating environment). That is, the effect of generation overlap is a "double-edged sword," with each edge cutting in a different direction. Here we use sexual (quantitative trait) and asexual (clonal) models to explore the effects of generation overlap on adaptive evolution in a fluctuating environment, either with or without a trend in the mean environment state. Our analyses show that when environmental stochasticity scaled by strength of selection is intermediate and when the trend in mean environment is slow, intermediate values of generation overlap can maximize the rate of response to selection and minimize the adaptation lag between the trait mean and the environmental trend. Otherwise, increased generation overlap results in smaller selection response and larger adaptation lag. In the former case, low generation overlap results in low heritable trait variance, while high generation overlap increases the "migration load" from the past. Therefore, to understand the importance of rapid evolution for organisms with overlapping generations, we need to understand the interaction of generation overlap, environmental stochasticity, and strength of selection.