You can plant container-grown plants any time of the year, unless the ground is frozen. If you take care to dig the correct size hole and minimize the disturbance to the plant's root ball in the container, the plant will hardly notice you have done anything to it. Make sure it has enough water after planting, until the roots start to grow out into the surrounding soil.
For other plants, as a general rule the best times are at the end of the growing season (for example when ornamental grasses start to die down, or deciduous trees start to drop their leaves) or in spring just before they restart growth. Obviously translating that advice into calendar dates depends on your climate!
If you transplant in autumn, the roots will continue growing in the soil which is still warm, before winter sets in. If you transplant "late" in spring, there will be less risk of disease and weather damage because the ground has been disturbed while the plant is not growing.
If you dig up an actively growing plant, the damage and disturbance to the roots reduces its ability to taking up water in the short term, and if you do that at the start of a summer heatwave the result could be fatal to the plant. An unseasonably cold and wet summer is actually good weather for redesigning your garden planting scheme!