The age old question of soil conditions before transplant. This will be determined more by what you want to accomplish from the step up than a hard and fast rule. First, daytime or nighttime is immaterial. For ease of removing the plant from the pot drier soil is the easiest, especially if you are newer to doing this. The drawback is if you want to work the roots, the drier the soil the more damage possible when scaring the roots. Wet soil is great if you plan on bare rooting your plant into the new container. When you are comfortable with transplanting this method gives you some great advantages over just sticking a plant into the new container.
When transplanting the roots should be disturbed, scared or something similiar to promote new root growth and to expand into the new container. Dry soil a knife run down the sides of the soil will do this. You need several days for the plant to fully recover in some instances and in other the plant will show no signs of stress from this. The disadvantage of the dry soil here is that the soil in the original container will not totally meld with the new. Bare rooting the transplant will give you great root growth faster, the soil is all together as one unit, important if you are an outside grower with wind issues. I say this but it really takes some practice to do this method the first few times but then becomes second nature.
So in essence the best time to transplant is when your roots come out of the bottom of the container to ensure it holds the soil properly, no specific time of day really although early morning I find makes for the healthiest transplant. IMHO this is because the roots have just finished their cycle of growth and the plant is fully charged for the day.
Good Luck