Isolated populations with novel phenotypes present an exciting opportunity to uncover the genetic basis of ecologically significant adaptation, and genomic scans have often, but not always, led to candidate genes directly related to an adaptive phenotype. However, in many cases these populations were established by a severe bottleneck, which can make identifying targets of selection problematic. Here, we simulate severe bottlenecks and subsequent selection on standing variation, mimicking adaptation after establishment of a new small population, such as an island or an artificial selection experiment. Using simulations of single loci under positive selection and population genetics theory, we examine how population size and age of the population isolate affect the ability of outlier scans for selection to identify adaptive alleles using both single-site measures and haplotype structure. We find and explain an optimal combination of selection strength, starting frequency, and age of the adaptive allele, which we refer to as a Goldilocks zone, where adaptation is likely to occur and yet the adaptive variants are most likely to derive from a single ancestor (a ‘hard’ selective sweep); in this zone, four commonly used statistics detect selection with high power. Real-world examples of both island colonization and experimental evolution studies are discussed. Our study provides concrete considerations to be made before embarking on whole-genome sequencing of differentiated populations.