We study how rugged fitness landscapes are explored by spatially structured populations with demes on the nodes of a graph, connected by migrations. In the weak mutation and rare migration regime, we find that, in most landscapes, migration asymmetries associated with some suppression of natural selection allow the population to reach higher fitness peaks first. In this sense, suppression of selection can make early adaptation more efficient. However, the time it takes to reach the first fitness peak is then increased. We also find that suppression of selection tends to enhance finite-size effects. Finite structures can adapt more efficiently than very large ones, especially in high-dimensional fitness landscapes. We extend our study to frequent migrations, suggesting that our conclusions hold in this regime. We then investigate the impact of spatial structure with rare migrations on long-term evolution by studying the steady state of the population with weak mutation, and introducing an associated steady-state effective population size. We find that suppression of selection is associated to small steady-state effective population sizes and thus to small average steady-state fitnesses.