Genetic spaces are often described in terms of fitness landscapes or genotype-to- phenotype maps, where each potential genetic sequence is associated with a set of properties and connected to other genotypes that are a single mutation away. The positions close to a genotype make up its mutational landscape and, in aggregate, determine the short-term evolutionary potential of a population. Populations with wider ranges of phenotypes in their mutational neighborhood tend to be more evolvable. Likewise, those with fewer phenotypic changes available in their local neighborhoods are more mutationally robust. As such, forces that alter the distribution of phenotypes available by mutation can have a profound effect on subsequent evolutionary dynamics. We demonstrate that cyclically-changing environments can push populations toward more evolvable mutational landscapes where a wide range of alternate phenotypes are available, though purely deleterious mutations remain suppressed. We further show that populations in environments with drastic changes shift phenotypes more readily than those in environments with more benign changes. We trace this effect to repeated population bottlenecks in the harsh environments, which result in shorter coalescence times and keep populations in regions of the mutational landscape where the phenotypic shifts in question are more likely to occur.